Non-degeneracy of polynomial maps with respect to global ... · Orientador: Prof. Dr. Marcelo José Saia Coorientador: Prof. Dr. Carles Bivià-Ausina USP – São Carlos Agosto de

Non-degeneracy of polynomial maps with

respect to global Newton polyhedra

Jorge Alberto Coripaco Huarcaya

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito: Assinatura:_______________________


Não-degeneração de aplicações polinomiais com respeito à poliedros de Newton globais

Tese apresentada ao Instituto de Ciências Matemáticas e de Computação - ICMC-USP, como parte dos requisitos para obtenção do título de Doutor em Ciências – Matemática. VERSÃO REVISADA

Área de Concentração: Matemática

Orientador: Prof. Dr. Marcelo José Saia

Coorientador: Prof. Dr. Carles Bivià-Ausina

USP – São Carlos Agosto de 2015

Este trabalho teve suporte financeiro da FAPESP, processos nº 2011/10653-6 e 2012/22365-8

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP,

com os dados fornecidos pelo(a) autor(a)

C798nCoripaco Huarcaya, Jorge Alberto Non-degeneracy of polynomial maps with respectto global Newton polyhedra / Jorge Alberto Coripaco Huarcaya; orientador Marcelo José Saia; co-orientador Carles Bivià-Ausina. -- São Carlos, 2015. 106 p.

Tese (Doutorado - Programa de Pós-Graduação emMatemática) -- Instituto de Ciências Matemáticas ede Computação, Universidade de São Paulo, 2015.

1. Non-degeneracy of polynomial maps. 2.Lojasiewicz exponent at infinity. 3. Global Newtonpolyhedra. 4. Special monomials. I. Saia, MarceloJosé, orient. II. Bivià-Ausina, Carles, co-orient.III. Título.


Non-degeneracy of polinomial maps with respect to global Newton polyhedra

Doctoral dissertation submitted to the Instituto de Ciências Matemáticas e de Computação - ICMC-USP, in partial fulfillment of the requirements for the degree of the Doctoral Program in Mathematics. FINAL VERSION

Concentration Area: Mathematics

Advisor: Prof. Dr. Marcelo José Saia

Coadvisor: Prof. Dr. Carles Bivià-Ausina

USP – São Carlos August 2015

This work has been partially supported by FAPESP, processes nº 2011/10653-6 and 2012/22365-8

Aos meus queridos pais, Juan de Dios e Alejandra.

Agradecimentos

Primeiramente eu quero agradeçer a Deus, por sempre guiar meu caminho e pela forçaque sempre me deu para alcançar meus objetivos.

Eu gostaria antes de mais nada de expressar meus mais sinceros agradecimentos a meusProfessores orientadores, Prof. Dr. Marcelo José Saia, pela sua orientação, apoio total,disponibilidade, amizade e pela confiança que sempre me deu ao longo da realização destetrabalho. Ao Prof. Dr. Carles Bivià-Ausina, pelo apoio durante minha estadia na Espanha,pela orientação, pelo saber que transmitiu, pela disponibilidade, atenção dispensada, paciên-cia, dedicação e profissionalismo pelos quais estou muito agradecido!

Agradeço também ao Prof. Dr. José Bonet, director do Instituto Universitario deMatemática Pura y Aplicada, Universidad Politécnica de Valencia, pela hospitalidade e asexcelentes condicões de trabalho que me ofereceu este instituto.

Agradeço à meus queridos pais, Juan de Dios Curipaco e Alejandra Huarcaya que sempreme deram seu amor e carinho, pelo apoio incondicional e todo o sacrifício que fizeram paraque eu continue meus estudos. À meus irmãos Moises, Duly, Betzabe, Efrain, Jose, Juan,Mayra e Issac pela amizade e pelo apoio durante toda minha vida.

Agradeço aos professores do departamento de matemática do ICMC-USP (São Carlos-Brasil), pelo ensinamentos, apoio e motivação academica.

Agradeço a FAPESP, Fundação de Amparo à Pesquisa do Estado de São Paulo pelo apoiofinanceiro, processos n0 2011/10653-6 e 2012/22365-8.

Finalmente agradeço a todas as pessoas que direta ou indiretamente participaram narealização deste trabalho.

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Abstract

Let F : Kn → Kp be a polynomial map, where K = R or C. Motivated by the characterizationof the integral closure of ideals in the ring On by means of analytic inequalities proven byLejeune-Teissier [46], we define the set Sp(F ) of special polynomials with respect to F . Theset Sp(F ) can be considered as a counterpart, in the context of polynomial maps Kn → Kp, ofthe notion of integral closure of ideals in the ring of analytic function germs (Kn, 0)→ K. Inthis work, we are mainly interested in the determination of the convex region S0(F ) formedby the exponents of the special monomials with respect to F .

Let us fix a convenient Newton polyhedron Γ+ ⊆ Rn. We obtain an approximation toS0(F ) when F is strongly adapted to Γ+, which is a condition expressed in terms of the facesof Γ+ and the principal parts at infinity of F . The local version of this problem has beenstudied by Bivià-Ausina [4] and Saia [71]. Our result about the estimation of S0(F ) allows usto give a lower estimate for the Łojasiewicz exponent at infinity of a given polynomial mapwith compact zero set. As a consequence of our study of Łojasiewicz exponents at infinity wehave also obtained a result about the uniformity of the Łojasiewicz exponent in deformationsof polynomial maps Kn → Kp. Consequently we derive a result about the invariance of theglobal index of real polynomial maps Rn → Rn.

As particular cases of the condition of F being adapted to Γ+ there appears the class ofNewton non-degenerate polynomial maps at infinity and pre-weighted homogeneous maps.The first class of maps constitute a natural extension for maps of the Newton non-degeneracycondition introduced by Kouchnirenko for polynomial functions. We characterize the Newtonnon-degeneracy at infinity condition of a given polynomial map F : Kn → Kp in terms ofthe set S0((F, 1)), where (F, 1) : Kn → Kp+1 is the polynomial map whose last componentfunction equals 1. Motivated by analogous problems in local algebra we also derive someresults concerning the multiplicity of F .Key words: Newton polyhedra, non-degeneracy conditions, multiplicity of polynomial maps,Łojasiewicz exponent at infinity, global injectivity of polynomial maps, index of real polyno-mials maps.

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Resumo

Seja F : Kn → Kp uma aplicação polinomial, onde K = C ou K = R. Motivados pelacaracterização do fecho integral de ideais no anel On por meio de desigualdades analíticasprovadas por Lejeune-Teissier [46], definimos o conjunto Sp(F ) de polinomios especiais comrespeito a F . O conjunto Sp(F ) pode ser considerado como um homólogo, no contexto dasaplicações polinomiais Kn → Kp, da noção de fecho integral de ideais no anel de germesde funções analíticas (Kn, 0) → K. Neste trablaho, estamos interessados principalmente nadeterminação da região convexa S0(F ) formado pelos expoentes dos monômios especiais comrespeito a F .

Fixado um poliedro de Newton conveniente Γ+ ⊆ Rn, é obtida uma aproximação de S0(F ),quando F é fortemente adaptada a Γ+ o qual é uma condição expressada em termos das facesde Γ+ e as partes principais no infinito de F . A versão local deste problema foi estudado porBivià-Ausina [4] e Saia [71]. Nosso resultado sobre a estimativa de S0(F ) nos permite daruma estimativa inferior para o expoente Łojasiewicz no infinito de uma aplicação polinomialKn → Kp, com conjunto F−1(0) compacto. Como uma consequência do estudo dos expoentesde Łojasiewicz no infinito também foi obtido um resultado sobre a uniformidade do expoenteŁojasiewicz em deformações de aplicações polinomiais Kn → Kp e consequentemente, umresultado sobre a invariância do índice global de aplicações polinomiais reais Rn → Rn.

Como casos particulares da condição de F ser adaptada a Γ+ aparecem a classe de apli-cações polinomiais Newton não degeneradas e as aplicações polinomiais pre-quase homoge-neas. A primeira classe de aplicações constitui uma extensão natural da condição Newtonnão-degeneração introduzida por Kouchnirenko para funções polinomiais. Caracterizamos acondição Newton não-degeneração para uma determinada aplicação polinomial F : Kn → Kp

em termos do conjunto S0((F, 1)), onde (F, 1) : Kn → Kp+1 é a aplicação polinomial cujaúltima função componente é igual a 1. Motivados por problemas análogos em álgebra local,também obtivemos alguns resultados sobre a multiplicidade de F .Palavras-chave: Poliedros de Newton, condições de não-degeneração, multiplicidade de aplicações

polinomiais, injectividade global de aplicações polinomiais, indice de aplicações polinomiais reais.

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Contents

Introduction 1

1 Preliminaries 51.1 Multiplicity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Newton non-degenerate ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Adapted systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Index theory and local algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Finite polynomial maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 Łojasiewicz exponent at infinity of polynomial maps . . . . . . . . . . . . . . . 19

2 Newton filtrations and non-degeneracy conditions 252.1 Global Newton polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Newton non-degeneracy at infinity . . . . . . . . . . . . . . . . . . . . . . . . 302.3 The Newton filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Counting affine roots of polynomial systems . . . . . . . . . . . . . . . . . . . 38

3 Newton non-degeneracy at infinity 413.1 Special monomials and Newton non-degeneracy at infinity . . . . . . . . . . . 423.2 Multiplicity and special polynomials . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Non-degeneracy with respect to a Newton filtration . . . . . . . . . . . . . . . 553.4 Newton non-degeneracy of gradient maps . . . . . . . . . . . . . . . . . . . . . 603.5 Homogeneous Newton polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Łojasiewicz exponent and adapted maps to Newton polyhedra 714.1 Maps adapted to Newton polyhedra . . . . . . . . . . . . . . . . . . . . . . . 724.2 Special monomials and adapted maps . . . . . . . . . . . . . . . . . . . . . . 794.3 Estimation of Łojasiewicz exponents at infinity . . . . . . . . . . . . . . . . . . 86

vii

4.4 Łojasiewicz exponent at infinity of pre-weighted homogeneous maps . . . . . . 894.5 Index of polynomial maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Introduction

Newton polyhedra have been used in different problems of local singularity theory, ascan be seen in the works of Bivià-Ausina [4], Fukui [32], Kouchnirenko [43], Oka [51], Saia[70], Wall [78] and Yoshinaga [79] among other authors. In [43], Kouchnirenko showed aformula to compute the Milnor number of a germ of analytic function f : (Cn, 0) → (C, 0)with isolated singularity at the origin in terms of the Newton polyhedron of f . Bivià-Ausina,Fukui and Saia in [11] characterized a class of ideals of finite codimension in the ring of formalpower series, which is expressed in terms of a fixed Newton polyhedra. This class includesNewton non-degenerate ideals in the sense of Saia [70] and ideals generated by semi-weightedpolynomials (with respect to a fixed vector of weights).

Concerning the polynomial case, Kouchnirenko also showed in [43] a formula for com-puting the global Milnor number of polynomials f ∈ C[x1, . . . , xn] with a finite number ofsingularities, under the condition that f is Newton non-degenerate at infinity. Newton poly-hedra were also used by Bivià-Ausina in [8] to obtain a lower estimate for the Łojasiewiczexponents at infinity of real polynomial maps.

These results are the main motivation of our work in order to apply global Newton polyhe-dra to study polynomial maps Kn → Kp, K = R or C. The subject of this thesis is the study ofof non-degeneracy conditions for polynomials maps Kn → Kp, with respect to global Newtonpolyhedra, in order to obtain polynomial versions of results described in [4, 6, 10, 11, 33, 43]and [70].

Motivated by the characterization given by Lejeune-Teissier [46], by means of analyticinequalities, of the integral closure of ideals in the ring On of germs of analytic functions(Cn, 0) → C, we introduce in Section 3.1.8 a version for polynomial maps of the notionof integral element, which we call special element with respect to a given polynomial mapsKn → Kp. This notion is fundamental in the development of this work. If F : Kn → Kp is apolynomial map, then we denote by Sp(F ) the set of special polynomials with respect to F .

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2 Introduction

We can consider Sp(F ) as an analogous notion of the integral closure of ideals in the contextof polynomial maps.

The main result of this thesis is the determination of the convex region S0(F ) formed bythe exponents of the special monomials with respect to F using a global Newton polyhedron(see Theorems 4.2.3 and 4.2.4). We obtain an approximation to S0(F ) when F is stronglyadapted to Γ+, which is a condition expressed in terms of the faces of the Newton polyhedronand the principal parts at infinity of F . We remark that the local version of this problem,has been studied previously by Bivià-Ausina in [4].

In this context, a very interesting problem is to give a lower estimate of the Łojasiewiczexponent at infinity for a polynomial map F : Kn → Kp, where K = R or C using globalNewton polyhedra. This number, which is denoted by L∞(F ), is defined as the supremum ofthose real numbers α such that there exist constants C,M > 0 such that

‖x‖α 6 C‖F (x)‖, (1)

for all x ∈ Kn such that ‖x‖ > M . The number L∞(F ) exists if and only if F−1(0) isa compact set. In this case, this is a rational number, which can be negative (see [44] fordetails). The Łojasiewicz exponent at infinity is intimately related with questions aboutinjectivity of polynomial maps [22] and the equivalence at infinity of polynomial vector fields[62].

The exact computation or the estimation of L∞(F ) from below is a non-trivial problem[25], [44], [60] and, in general, there are no explicit formulas for the Łojasiewicz exponent atinfinity.

As a consequence of our result on the estimation of S0(F ), we obtain a lower estimatefor Łojasiewicz exponent at infinity of real and complex polynomial maps (see Corollary4.3.3 and Proposition 4.4.2). Płoski [60], Kollár and Shiffman [69] and Cygan, Krasińskiand Tworzewski [25] also gave of lower estimates for the Łojasiewicz exponent at infinity forcomplex polynomial maps, using different methods.

Moreover, from our results about the estimation of Łojasiewicz exponents at infinity weshow a result concerning the global index of a given polynomial map F : Rn → Rn (seeTheorem 4.5.1) with finite zero set. This result describes which monomials can be added toeach component function of F leading to a map having the same global index than F . Thelocal version of this question, which is analyzed in the articles [5, 19, 36] and [48], takes partof the wider problem of determining which monomials in the Taylor expansion of a smoothvector field determine the local phase portrait (see for instance [14, 16] and [80]). The first

Introduction 3

step in this approach to the study of global indices is the result of Cima-Gasull-Mañosas [18,Proposition 2] on the index of maps whose monomials of maximum degree with respect tosome vector of weights have an isolated zero. We call these maps pre-weighted homogeneous(see Definition 2.3.5 for a precise formulation of this concept).

This thesis is organized as follows:

Chapter 1. In Section 1.1 we recall several basic notions from commutative algebra, likethe multiplicity of an ideal in a local ring, in the sense of Hilbert-Samuel, and the integralclosure of an ideal. Let On denote the ring of analytic function germs (Cn, 0)→ C, 0. Section1.2 contains some results shown in [11] for ideals in On; these results connect the notionsof multiplicity, integral closure and Newton non-degeneracy. In Section 1.3 we show thenotion of adapted system introduced by Bivià-Ausina in [4] and the result of [4] about theestimation of the Newton polyhedron of the monomials that belong to the integral closure ofideals I ⊆ On such that I admits a generating system which is adapted to a given Newtonpolyhedron. In Sections 1.4 and 1.5 we recall some preliminary results about the global indexof maps Rn → Rn (see for instance Theorem 1.4.7) and the multiplicity of a polynomial mapCn → Cn with finite zero set. Section 1.6 is devoted to showing known results about theŁojasiewicz exponent at infinity of a polynomial map Kn → Kp.

Chapter 2. In Section 2.1 we introduce the concept of global Newton polyhedra and ba-sic combinatorial results. In Section 2.2 we introduce the notion of Newton non-degeneracyat infinity for polynomial map Kn → Kp (see Definition 2.2.1) and recall the known resultof Kouchnirenko about the global Milnor number of a polynomial function. Section 2.3 isdevoted to introducing the Newton filtration associated to a convenient global Newton poly-hedron and the corresponding notion of non-degeneracy (see Definition 2.3.4) for polynomialmaps Kn → Kp. In Section 2.4 we recall a fundamental result of Rojas and Wang [67] aboutthe determination of the number of isolated affine roots, counting multiplicities, of polynomialmaps Cn → Cp using global Newton polyhedra.

Chapter 3. This chapter is devoted to showing analogous results for polynomial maps of theresults given in [11], [6] and [70]. In Section 3.1 we introduce the notion of special polynomialwith respect to a given polynomial map Kn → Kp and we show a characterization of theNewton non-degeneracy at infinity of polynomial maps in terms of this notion (see Theorem3.1.14). In particular, when F is a convenient polynomial map, we obtain the polynomialversion of a result shown by Saia [70, Theorem 3.4]. In Section 3.2 we show a necessary andsufficient condition for a polynomial to be special with respect to a given finite polynomial

4 Introduction

map Cn → Cn in terms of the notion of multiplicity (see Theorem 3.2.2). Moreover, wecharacterize the Newton non-degeneracy at infinity for polynomial maps F : Cn → Cn, interms of the set S0(F ) and the multiplicity of F (see Corollary 3.2.4). Section 3.3 containsversions for polynomial maps of some results proven by Bivià-Ausina, Fukui and Saia [11,Theorem 3.3] and Furuya and Tomari [33, Theorem 1]. In Section 3.4 we analyze the Newtonnon-degeneracy of gradient maps and relate this property with the Newton non-degeneracy ofpolynomial functions. Section 3.5 contains a characterization of an important class of globalNewton polyhedra.

Chapter 4. In this chapter, we approach the problem of determining when a monomial isspecial with respect to a given polynomial map using Newton polyhedra at infinity. For thispurpose, in Section 4.1, we introduce the concepts of adapted and strongly adapted mapwith respect to a fixed convenient Newton polyhedron at infinity. In Section 4.2 we showthe fundamental results about the approximation of the Newton polyhedron of the specialmonomials with respect to a given polynomial map (Theorems 4.2.3 and 4.2.4). In Sections4.3 and 4.4 we apply Theorem 4.2.4 to establish a positive lower bound for L∞(F ) (Corollary4.3.3 and Proposition 4.4.2) and to derive a consequence about the injectivity of polynomialmaps (see Corollary 4.3.6). Finally in Section 4.5 we apply the argument of the proof ofTheorem 4.2.4 to show a result about the invariance of global index of polynomial mapsRn → Rn.

CHAPTER 1

Preliminaries

Contents1.1 Multiplicity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Newton non-degenerate ideals . . . . . . . . . . . . . . . . . . . . . 91.3 Adapted systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Index theory and local algebra . . . . . . . . . . . . . . . . . . . . 161.5 Finite polynomial maps . . . . . . . . . . . . . . . . . . . . . . . . . 191.6 Łojasiewicz exponent at infinity of polynomial maps . . . . . . . 20

In this chapter, first we briefly recall the definitions of multiplicity in the sense of Samueland integral closure of ideals. Then, with the purpose of stating results shown by Bivià-Ausina, Fukui and Saia in [11], the condition of Newton non-degeneracy for ideals in thering of formal power series around the origin is introduced. In Section 1.3 it is shown aresult proved by Bivià-Ausina [4], that gives an estimation of the Newton polyhedron of themonomials that belong to the integral closure of an ideal.

1.1 Multiplicity theory

We begin this section by recalling the definition and some fundamental facts concerningthe multiplicity of an ideal, in the sense of Samuel, and the integral closure of ideals. Werefer to [15, 52] and [72] for more information about these concepts.

Theorem 1.1.1 ([72, Theorem 11.1.3]). Let (R,m) be a Noetherian local ring, let I be anm-primary ideal, and let M be a non-zero finitely generated R-module. Set d = dim(R).

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6 Chapter 1. Preliminaries

There exists a polynomial P (n), with rational coefficients, such that for all n 0,

P (n) = λR(M/InM),

where λR(M/InM) is the length of M/InM , as an R-module, for all n > 0. Furthermore, thedegree in n of P (n) is dimM , which is at most d. λR(M/InM) is called the Hilbert functionand P (n) the Hilbert–Samuel polynomial of I with respect to M . If the dependence of P (n)on I and M needs to be specified, then write PI,M(n).

Definition 1.1.2. Let (R,m) be a Noetherian local ring of dimension d, let I be an m-primaryideal, and let M be a finitely generated R-module. The multiplicity of I with respect to M ,denoted eR(I;M) is d! times the coefficient of PI,M(n) of degree d. If M = R, we simplywrite e(I) for eR(I;R), while if M = R and I = m, we write e(R) = e(m), the multiplicity ofR. In other words, e(I;M) = limn→∞ λ(M/InM)d!/nd, where d = dimR.

The multiplicity of an ideal can be interpreted geometrically and it is related to the notionof integral closure of an ideal.

Definition 1.1.3. Let I be an ideal in an arbitrary ring R and h ∈ R. An element h is integralover I when there is a monic polynomial of the form p(x) = xm + a1x

m−1 + · · ·+ am−1x+ am,with ai ∈ I i, m > 1, such that p(h) = 0. The subset of R of those elements which are integralover I is an ideal called the integral closure of I and is denoted by I. The ideal I contains Iand, if I = I, I is called integrally closed.

The next definition is an essential tool for the study of the integral closure of ideals.

Definition 1.1.4. Given a pair of ideals I and J in a Noetherian ring R such that J ⊆ I, Jis said to be a reduction of I when there exists an integer r > 0, such that Ir+1 = JIr.

We state the following result for Cohen-Macaulay rings, however, this result is true forformally equidimensional rings, which constitute a much wider class of rings (we refer to [52]and [72] for details).

Theorem 1.1.5 (Rees’ Theorem [63]). Let (R,m) be a regular local ring, and let J and I be apair of m-primary ideals of R such that J ⊆ I. Then the following conditions are equivalent:

(i) J is a reduction of I;

(ii) I = J ;

(iii) e(I) = e(J).

Chapter 1. Preliminaries 7

The equivalency between (i) and (ii) is true when I and J are not m-primary ideals, as isshown in [72, Corollary 1.2.5].

Next we give a result from [52, Theorem 14.14] on the existence of reductions of an ideal Iin a local ring, see also [58, p. 112]. As we see, reductions of an ideal can be obtained throughgeneric linear combinations of a given generating system of I.

Theorem 1.1.6. Let R be an n-dimensional local ring whose residual field is equal to k.Let us assume that k is infinite. Let I ⊆ R be an ideal of finite colength. Suppose that Iis generated by g1, . . . , gs. Then there exists an analytic set S(I) ⊆ ks × kn such that, if(a11, . . . , a1s, . . . , an1, . . . , ans) ∈ ks × kn \ S(I) and we define hi = ∑

j aijgj, i = 1, . . . n, thenthe ideal of R generated by h1, . . . , hn is a reduction of I.

Let On denote the ring of analytic function germs (Cn, 0) → C. Next, we state a resultof Lejeune-Teissier [46] characterizing the integral closure of ideals in On.

Theorem 1.1.7. Let I be an ideal in On. The following statements are equivalent:

(i) h ∈ I.

(ii) (Growth condition) For each system of generators g1, . . . , gs of I there exists a neigh-borhood U of 0 and a constant C > 0 such that |h(x)| 6 C sup|gi(x)| : i = 1, . . . , s,for all x ∈ U .

(iii) (Valuative criterion) For each analytic curve ϕ : (C, 0)→ (Cn, 0), the germ hϕ lies in(ϕ∗(I))O1, where ϕ∗ : On → O1 denotes the natural morphism induced by compositionby ϕ.

In the following section, we will see some results about the computation of the multiplicityand the integral closure of ideals.

1.2 Newton non-degenerate ideals

Let us fix an integer n > 1 and let A(Kn) denote the ring of analytic function germs(Kn, 0) → K, where K = R or C. We usually write On to denote A(Cn). We recall thatA(Kn) is a local ring whose maximal ideal, that we denote by mn (or simply, by m if no riskof confusion arises) is formed by the germs h ∈ A(Kn) such that h(0) = 0. In this section weintroduce the condition of Newton non-degeneracy for ideals in A(Kn).

Let us fix coordinates x1, . . . , xn in Kn. We set R>0 = [0,+∞[, Z>0 = Z ∩ R>0 andQ>0 = Q ∩ R>0. If k = (k1, . . . , kn) ∈ Zn>0, then we denote the monomial xk1

1 · · ·xknn by xk.


Let g = ∑akx

k be an element of A(Kn), the support of g, denoted by supp(g), is define asthe set of points k ∈ Zn such that ak 6= 0. If I is an ideal of A(Kn), we define the support ofI as supp (I) = ∪g∈I supp (g).

Definition 1.2.1. A subset Γ+ of Rn>0 is a Newton polyhedron, or a local Newton polyhedron,

when there exists some A ⊆ Qn>0 such that Γ+ is equal to the convex hull in Rn

>0 of the setk + v : k ∈ A, v ∈ Rn

>0. In this case we say that Γ+ is the Newton polyhedron determinedby A and we also denote it by Γ+(A).

Let g ∈ A(Kn), then the Newton polyhedron of g is defined as Γ+(g) = Γ+(supp(g)). LetI be an ideal of A(Kn), then the Newton polyhedron of I, denoted by Γ+(I), is defined asΓ+(supp(I)). We observe that if g1, . . . , gs is a generating system of I ⊆ A(Kn), then Γ+(I)is equal to the convex hull of Γ+(g1) ∪ · · · ∪ Γ+(gs). It is also obvious that, if J ⊆ I are twoideals of A(Kn), then Γ+(J) ⊆ Γ+(I).

Let Γ+ ⊆ Rn>0 be a Newton polyhedron and let v ∈ Rn

>0. We define

`(v,Γ+) = min〈v, k〉 : k ∈ Γ+; (1.1)

∆(v,Γ+) = k ∈ Γ+ : 〈v, k〉 = `(v,Γ+). (1.2)

A face of Γ+ is any set of the form ∆(v,Γ+) for some v ∈ Rn>0. The dimension of a face ∆

of Γ+, denoted by dim ∆, is defined as the dimension of the smallest affine subspace of Rn>0

containing ∆. We denote by Γ the union of the compact faces of Γ+ and by Γ− the union ofall segments joining the origin and a point of Γ. If I is an ideal of A(Kn), then the sets Γ(I)and Γ−(I) are defined analogously. We denote the n-dimensional volume of Γ−(I) by Vn(I).

If I is an ideal of A(Kn) of finite colength, then Γ+(I) intersects all coordinate axes, sinceI contains a power of the maximal ideal of A(Kn). In this case we observe that Γ−(I) is equalto the closure (in the Euclidean sense) of the set Rn

>0 \ Γ+(I).Let ∆ be a compact face of a Newton polyhedron Γ+ and let C(∆) denote the cone

obtained by all half lines starting at the origin and passing through some point of ∆. Thenwe denote by A∆ the subring of On given by all h ∈ On such that supp(h) ⊆ C(∆). Weremark that A∆ is a local Cohen-Macaulay ring of dimension dim(∆) + 1 (see [43, p. 24 ]).

If g = ∑k akx

k ∈ On, we denote by g∆ the polynomial obtained by the sum of all termsakx

k such that k ∈ supp(g) ∩∆. If supp(g) ∩∆ = ∅, then we set g∆ = 0.

Definition 1.2.2 ([70, p. 2]). Let I be an ideal of On and let g1, . . . , gs be a generatingsystem of On. Then I is Newton non-degenerate if, for each compact face ∆ of Γ+(I), theideal generated by (g1)∆, . . . , (gs)∆ in A∆ has finite colength in A∆.


Remark 1.2.3. We recall that it is immediate to see that the above definition does not de-pend on the given generating system of I. The fact that (g1)∆, . . . , (gs)∆ has finite colengthin A∆ is equivalent to saying that, for all face ∆1 ⊆ ∆, the set of common zeros of(g1)∆1 , . . . , (gs)∆1 is contained in x ∈ Cn : x1 · · ·xn = 0 (see [43, Theorem 6.2]).

The above observation motivates the following definition. If I is an ideal of A(Rn) andg1, . . . , gs denotes a generating system of I, then we say that I is Newton non-degeneratewhen the set of common zeros of (g1)∆1 , . . . , (gs)∆1 is contained in x ∈ Rn : x1 · · ·xn = 0.

If I is an ideal in A(Kn), then denote by I0 the ideal of A(Kn) generated by those monomialsxk such that k belongs to Γ+(I).

Lemma 1.2.4 ([11, Lemma 2.3]). Let I be an ideal of A(Kn), then Γ+(I) = Γ+(I).

Let I be an ideal of A(Kn). The set C(I) is defined as the convex hull in Rn>0 of k ∈

Zn>0 : xk ∈ I. That is, C(I) is the Newton polyhedron of the ideal KI generated by all themonomials belonging to the integral closure of I. The next result is shown for ideals of finitecodimension in On in [70], but the same proof works for ideals of A(Kn), with no additionalconditions.

Theorem 1.2.5 ([70, Theorem 3.4]). Let I be an ideal of On of finite codimension, thenC(I) ⊆ Γ+(I). The equality Γ+(I) = C(I) holds if and only if I is Newton non-degenerate.

The following result characterizes the reductions of Newton non-degenerate ideals and itis a consequence of Lemma 1.2.4, Theorem 1.2.5 and the Rees’ Multiplicity Theorem.

Corollary 1.2.6. Let I be a Newton non-degenerate ideal of On. Let J ⊆ I be another ideal ofOn. Then J is a reduction of I if and only if J is Newton non-degenerate and Γ+(I) = Γ+(J).

The following result appears in [11] and includes a characterization of the Newton non-degeneracy property in terms of the value of the Samuel multiplicity.

Theorem 1.2.7. Let I ⊆ On be an ideal of finite codimension. Then, the following conditionsare equivalent:

(i) I is a Newton non-degenerate ideal,

(ii) Γ+(I) = C(I),

(iii) e(I) = n!Vn(I),

(iv) I is generated by monomials,


(v) I0 ⊆ I,

(vi) I = f ∈ On : Γ+(f) ⊆ Γ+(I).

The above result constitutes a motivation to investigate the characterization of Newtonnon-degeneracy at infinity for polynomial maps Kn → Kn, which is a notion defined in thenext chapter.

In [6] Bivià-Ausina generalized the above result for a class of ideals that do not have finitecodimension in general. We recall some definitions in order to introduce this kind of ideals.

Definition 1.2.8. Let I be an ideal of On, we set

Λ+(I) = λk : λ > 0, k ∈ Γ(I) ; (1.3)

AI = h ∈ On : supp(h) ⊆ Λ+(I) ; (1.4)

c(I) = max dim ∆ : ∆ is a compact face of Γ(I) . (1.5)

We observe that, in general, the set Λ+(I) is not convex in Rn>0. Nevertheless, it is not

the case when n = 2. If Λ+(I) is convex, then Λ+(I) ∩ Zn>0 is a sub-semigroup of Zn>0 andAI becomes a subring of On. In fact, AI is a local ring of dimension c(I) + 1 in this case.It is said that I has finite relative colength when Λ+(I) is convex and I admits a generatingsystem g1, . . . , gs such that supp(g1) ∪ · · · ∪ supp(gs) ⊆ Λ+(I), and the ideal in AI generatedby g1, . . . , gs has finite colength in AI .We observe that Γ−(I) is contained in an affine subspace of dimension c(I) + 1. Then we canconsider the (c(I) + 1)-dimensional volume of Γ−(I); we denote this volume by v(I).

When I is an ideal of On of finite colength, then I has finite relative colength since theNewton polyhedron of I intersects all the coordinate axes and consequently AI = On in thiscase.

If I ⊆ On is an ideal of finite relative colength, we can consider I as an ideal of AI .Therefore, it makes sense to compute the multiplicity of I in AI since AI is a local ring. Thismultiplicity is denoted by e(I,AI). In the sequel, I1 denotes the ideal of AI generated by allthe monomials xk, with k ∈ Λ+(I) ∩ Zn>0 (compare with the definition of the ideal I0).

Theorem 1.2.9 ([6, Theorem 4.3]). Let I ⊆ On be an ideal of finite relative colength. ThenI is Newton non-degenerate if and only if e(I,AI) = (c(I) + 1)!v(I).

In the remaining section we recall the definition of a class of ideals of On studied in [11] byBivià-Ausina, Fukui and and Saia that strictly contains the class of Newton non-degenerateideals generated by n elements.


Let us consider Γ+ ⊆ Rn>0 a Newton polyhedron at the origin, such that Γ+ intersects all

coordinate axes. From the boundary of the Newton polyhedron Γ+ ⊆ Rn we can construct apiecewise linear function ΦΓ : Rn → R with the following properties:

(i) ΦΓ is linear on each cone C(∆), where ∆ is a compact face of Γ+.

(ii) ΦΓ takes positive integer values on the points of Zn>0.

(iii) there exists a positive integer M such that ΦΓ(k) = M , for all k ∈ Γ+.

When the map ΦΓ is constructed we define the ideals

Aq =g ∈ On : supp (g) ⊆ Φ−1

Γ (q + N), for all q > 0;

thenOn = A0 ⊇ A1 ⊇ A2 ⊇ . . .

is a filtration of ideals of On, that is ApAq ⊆ Ap+q, for all p, q ∈ Z>1. This is called theNewton filtration of On associated to Γ+ (see also [43, 2.1]). For any compact face ∆ of Γ+,this filtration induces a filtration on A∆ in a natural way.

Let g = ∑akx

k ∈ On, then the level of g with respect to the filtration given above isdefined as

νΓ(g) = min ΦΓ(k) : k ∈ supp(g) = max q : g ∈ Aq

The principal part of g, denoted by in(g), is the sum of terms akxk such that ΦΓ(k) = νΓ(g).Given a face ∆ of Γ+, the principal part of g over ∆, denoted by in∆(g), is the polynomialgiven by the sum of terms akxk such that ΦΓ(k) = νΓ(g) and k ∈ supp(g) ∩ C(∆).

Definition 1.2.10. Let I be an ideal of finite colength of On. A system of generatorsg1, . . . , gs of I is non-degenerate on Γ+ if, for each compact face ∆ ⊆ Γ, the ideal of A∆

generated by in∆(g1), . . . , in∆(gs) has finite colength in A∆. When the system g1, . . . , gs doesnot satisfy the above definition, we say that this is degenerate on Γ+.

Remark 1.2.11. Analogously to Definition 1.2.2, the ideal of A∆ generated by the systemin∆(g1), . . . , in∆(gs) has finite colengh in A∆ if and only if, for each compact face ∆1 ⊆ ∆,the set of common zeros of in∆1(g1), . . . , in∆1(gs) is contained in x ∈ Cn : x1 · · ·xn = 0

Next, we have the main result of a non-degenerate system on Newton polyhedron, thatgives us a tool to calculate the codimension of ideals in On. More details about of non-degenerate system on Newton polyhedron, can be found in [11].


Theorem 1.2.12 ([11, Theorem 3.3]). Let g1, . . . , gn be a system of generators of an ideal Iwith finite codimension in On and let Γ+ ⊆ Rn be a Newton polyhedron. If M is the value onΓ of the filtration induced by Γ+ and d1 = νΓ(g1), . . . , dn = νΓ(gn) are the levels of the givenset of generators of I with respect to this filtration, then

dimCOn/I > (d1 · · · dn/Mn)n!Vn(Γ−),

and equality holds if and only if the system g1, . . . , gn is non-degenerate on Γ+.

In Chapter 3 we will give a polynomial version of this result.

1.3 Adapted systems

Let Γ+ denote a convenient Newton polyhedron in Rn. In this section we recall a resultfrom [4] that, in particular, gives an estimation of the Newton polyhedron of the monomialsthat belong to the integral closure of I.

If S ⊆ A(Kn), where K = R or C. We denote by V (S) the zero set germ at 0 of S in Kn.

Definition 1.3.1. Let I be an ideal in A(Kn) and h ∈ A(Kn) such that V (I) ⊆ V (h). Letg1, . . . , gs be a system of generators of I. By [50, p. 136] (see also [13, p. 493]), we canconsider the greatest lower bound of those θ > 0 such that there exists an open neighborhoodU of 0 in Kn and a constant C > 0 such that

|h(x)|θ 6 C supi|gi(x)|,

for all x ∈ U . This number is called the Łojasiewicz exponent of h with respect to I and isdenoted by `(h, I). It is easy to see that this definition does not depend on the chosen systemof generators of I.

Lejeune-Tessier showed in [46] that, if I is an ideal of A(Cn) then, `(h, I) is a rationalnumber and Bochnak-Risler in [13] showed that this is also true when I is an ideal of A(Rn).

If I is an ideal of A(Rn), then the integral closure of I, denoted by I, is defined as theideal of A(Kn) generated by those h ∈ A(Rn) for which `(h, I) 6 1 (see [34]). ObviouslyI ⊆ I.

Moreover, if θ is a positive rational number, we denote by Iθ the ideal generated by thoseh ∈ A(Kn) such that `(h, I) 6 1/θ. Then `(h, I) can be expressed as

`(h, I) = 1maxθ ∈ Q>0 : h ∈ Iθ

.


The next result shows a sufficient condition for a monomial xk to belong to the ideal Iθ,where θ is a positive rational number. In particular, when θ = 1, then xk belongs to I.

Given a vector v ∈ Rn>0\0 and a series g = ∑

k akxk, then `(v, g) is defined as `(v,Γ+(g))

and ∆(v, g) = ∆(v,Γ+(g)) ∩ supp(g).If v ∈ Zn>0 \ 0, then we say that v is primitive when the non-zero coordinates of v

are mutually prime integers. In the remaining section, let us fix a local Newton polyhedronΓ+ ⊆ Rn. Suppose that the number of (n− 1)-dimensional faces of Γ+ is r. Let v1, . . . , vr bethe primitive vectors of Γ+ supporting some face of dimension n− 1 of Γ+.

Definition 1.3.2. Let g = ∑k akx

k and let J ⊆ 1, . . . , r, then we consider the set ∆(J, g) =∩j∈J∆(vj, g). The principal part of g with respect to J , denoted by qJ(g), is defined as thepolynomial given by the sum of those akxk such that k ∈ ∆(J, g). If ∆(J, g) = ∅, then we setqJ(g) = 0.

The following definition shows a concept that is the motivation for some ideas developedin Chapter 4.

Definition 1.3.3. Given S = g1, . . . , gs ⊆ A(Kn), then S is an adapted system to Γ+ (orS is K-adapted to Γ+) when, for each J ⊆ 1, . . . , r such that ∩j∈J∆(vj,Γ+) is a compactface of Γ+, the system of equations

qJ(g1)(x) = · · · = qJ(gs)(x) = 0 (1.6)

has no solution in (K∗)n. We say that S is an adapted system when S is adapted to theNewton polyhedron determined by S.

It is straightforward to check that, if S = g1, . . . , gs is non-degenerate with respect to aNewton polyhedron Γ+, then S is also adapted to Γ+ (in general, the converse is not true).

Theorem 1.3.4 ([4, Theorem 3.5]). Let Γ+ ⊆ Rn be a Newton polyhedron and let S =g1, . . . , gs be an adapted system to Γ+. Let I ⊆ A(Kn) be the ideal generated by S and letθ ∈ Q>0. Suppose that xk is a monomial such that

〈k, vj〉 > θmax`(vj, g1), . . . , `(vj, gs)

for all j = 1, . . . , r. Then xk ∈ Iθ.

In particular if S is a Newton non-degenerate system of A(Kn), the above result establishesthat the set k ∈ Zn>0 : xk ∈ I is equal to set Γ+(I) ∩ Zn>0.


1.4 Index theory and local algebra

In this section we describe some basic facts concerning the notion of index of real analyticmaps (Rn, 0)→ (Rn, 0). We refer to [26] for further information about this notion.

Along this section, we denote by Ω an open connected set in Cn. The mapping F =(F1, . . . , Fp) : Ω ⊆ Cn → Cp will be called a holomorphic mapping if the p functions F1, . . . , Fp

are holomorphic functions in Ω. If each Fi is a polynomial in the variables x1, . . . , xn, thenF is called a polynomial map.

If F : Ω ⊆ Cn → Cp is a holomorphic mapping, then we denote by FR : Ω ⊆ R2n → R2p

the real mapping obtained from F under the identification x + iy ↔ (x, y) between C andR2.

Lemma 1.4.1. Suppose that F : Ω ⊆ Cn → Cn is a holomorphic mapping. Then the Jacobiandeterminant of F satisfies the relation

det(dFR) = |det(dF )|2 (1.7)

As a consequence of this lemma, an holomorphic mapping has an invertible derivativeprecisely when the FR real mapping has an invertible derivative.

Definition 1.4.2. Suppose that F : Ω ⊆ Cn → Cp is a holomorphic mapping, where p > n.Then F is called a finite analytic mapping if, for each w ∈ F (Ω), the fiber F−1(w) is a finiteset.

Let us remark that proper holomorphic mappings between domains in complex Euclideanspaces are finite mappings. This follows because the inverse image of a point is necessarily acompact complex analytic subvariety.

Notation 1.4.3. Recall that we denote by F : (Cn, a)→ (Cp, 0), the germ of a holomorphicmapping from Cn to Cp such that F (a) = 0. A germ F : (Cn, a) → (Cp, 0) defines a finiteanalytic mapping when F−1(0) = a, as set germs. This means that there is a neighborhoodof a on which a is the unique solution to the equation F (z) = 0.

In order to study the index of an holomorphic mapping, we consider n = p. The indexis a generalization of the winding number of a curve in the complex plane. It can be usedto estimate the number of solutions of an equation, and is closely connected to fixed-point


theory. Let us suppose that a ∈ Ω is an isolated point of the set F−1(0). Let D be a ballcentered at a, with D ⊆ Ω, such that F−1(0) ∩D = a and let us consider the map

F

‖F‖: ∂D −→ Sn−1 (1.8)

where ∂D denotes the boundary of D. From the point of view of homology, we identify∂D with the unit sphere. Recall that the (n − 1)st homology group of Sn−1 is the group ofintegers, that is

Hn−1(Sn−1) ∼= Z. (1.9)

The continuous mapping (1.8) induces a mapping on the homology group Z. This inducedmapping is a group homomorphism and therefore must be multiplication by some integer.This integer is commonly called the topological degree of the mapping (1.8). Its value isindependent of the choice of sphere ∂D (see [26]).

Definition 1.4.4. Let F : (Rn, a)→ (Rn, 0) be the germ of a smooth mapping, and supposethat a is an isolated point in F−1(0). The index of F at the point a, written inda(F ), is thetopological degree of (1.8).

The simplest example about the computation of inda(F ) is when F is an orthogonal lineartransformation. Such a transformation preserves the sphere, and either preserves or reversesorientation. Hence deg(F ) = det(F ), thus ind0(F ) equals the determinant of F and can benegative in this case.

A version of the change of variables formula for multiple integrals holds for mappings withfinite topological degree. We state a simple version of that formula:

indx0(F )∫

‖x‖6ε

dV =∫

‖F (x)‖6ε

det(dF )dV, (1.10)

where ε denotes a small enough positive number. Here both integrals have the same orien-tation. This formula holds for mappings that are generically d-to-one, where d = indx0(F ),when the set of points with fewer than d inverse images has measure zero. Note also from(1.10) that det(dF ) cannot vanish identically.

We remark that for an holomorphic mapping, the index never cannot be negative. Thisfact follows from Lemma 1.4.1, which states essentially that a holomorphic mapping must beorientation preserving, and equality (1.10).

Corollary 1.4.5 ([26, p. 47]). The index of the germ of an holomorphic mapping F : (Cn, 0)→(Cn, 0) is positive.


Definition 1.4.6. Let Ω be an open subset of Cn and suppose that F : Ω ⊆ Cn → Cn isan holomorphic mapping such that F−1(0) is a finite set. The index of F in Ω denoted byind(F,Ω) is defined by

ind(F,Ω) =∑

a∈F−1(0)∩Ωinda(F )

If Ω = Cn, then we denote ind(F,Cn) by ind(F ).

The following result is well known, which is about the invariance of the index by homo-topies and can be found in [49, Theorem 2.2.4] and [18].

Theorem 1.4.7. Let F and G be continuous maps defined on the closure of a open connectedand bounded set Ω of Rn. Assume that F and G are homotopic on the boundary ∂Ω (thatis, there is a continuous homotopy H(t, z) : [0, 1] × Ω → Rn between F and G, such thatH(t, z) 6= 0 for all z ∈ ∂Ω). Then

ind(F,Ω) = ind(G,Ω).

As a consequence of the above theorem, we can consider the case when F has finitelymany roots aj inside a fixed ball B. Each root contributes to the topological degree of themapping F/‖F‖ : ∂B → S2n−1 between spheres, as described by the next proposition.

Proposition 1.4.8 ([26, p. 48]). Suppose B ⊆ Cn is a ball, and f : B → Cn is holomorphic,continuous on ∂B, and does not vanish there. If the mapping f/‖f‖ : ∂B → S2n−1 has degrees, then f has at most s roots in B and

s =∑

f(aj)=0indaj(f) (1.11)

Finally we have a result that allows us to calculate the index of an holomorphic mappingusing some tools of local algebra. If z ∈ Cn, then we denote by On,z the ring of analyticfunction germs (Cn, z)→ C.

Theorem 1.4.9 ([26, p. 60, Theorem 1]). Let F : Cn → Cn be an holomorphic mapping suchthat F−1(0) is a finite set. Then for each z ∈ F−1(0),

indz(F ) = dimCOn,z/Iz(F ),

where Iz(F ) is the ideal of On,z generated by the germs of the component functions of F at z.


1.5 Finite polynomial maps

Let I be an ideal of C[x1, . . . , xn], then we denote by V (I) the zero set of I. The followingtheorem characterizes a finite algebraic set.

Theorem 1.5.1 ([20, p. 43]). Let I be an ideal in C[x1, . . . , xn]. Then V (I) is a finite setif and only if C[x1, . . . , xn]/I is a finite dimensional vector space over C. If this occurs, thenumber of points in V (I) is at most dimC(C[x1, . . . , xn]/I).

Definition 1.5.2. Let I be an ideal in C[x1, . . . , xn]. Then, we say that I has finite codi-mension if C[x1, . . . , xn]/I is a finite dimensional vector space over C.

Let us observe from the above theorem that I has finite codimension if and only if V (I) isfinite.

Let I be an ideal in C[x1, . . . , xn] and let a a point in V (I). We consider the following set,

Sa := g ∈ C[x1, . . . , xn] : g(a) 6= 0 ,

which is a multiplicative set of R = C[x1, . . . , xn]. Let Ra := S−1a R be the quotient ring of R

by Sa (the localization of R at Sa), that is,

Ra := f/g : f, g ∈ C[x1, . . . , xn] and g(a) 6= 0 .

It is known that Ra is a local ring.

Definition 1.5.3. Suppose that a is a zero of I. The algebraic multiplicity of I at a, denotedby µa(I), is defined by

µa(I) = dimCRa/S−1a I.

In particular, if I is an ideal of C[x1, . . . , xn] of finite codimension, we have an relationamong the sum of the algebraic multiplicities of I and the codimension of I in C[x1, . . . , xn],as is shown in the following theorem.

Theorem 1.5.4 ([20, p. 150]). Let I be an ideal of C[x1, . . . , xn] with finite codimension.Then

dimC C[x1, . . . , xn]/I =∑

a∈V (I)µa(I) =

∑a∈V (I)

dimCOn,aIOn,a

.


Let F : Cn → Cn be a polynomial map, then denote by I(F ) the ideal of C[x1, . . . , xn]generated by the component functions of F . We say that F is a finite polynomial map when#F−1(0) <∞ (or equivalently, when I(F ) has finite codimension).

If F : Cn → Cn is a finite polynomial map, then the multiplicity of F , denote by µ(F ), isdefined as

dimCC[x1, . . . , xn]

I(F ) .

Let us observe from the above theorem that F is finite if and only if µ(F ) <∞.

Recall that a quasi-affine variety in Cn is a Zariski-open subset of an affine variety in Cn (see[38, 73] and [21],). A constructible set is a finite union of quasi-affine varieties. Constructiblesets are closed under complementation, projection, and finite union and intersection. Notethat a dense constructible set has nonempty interior.

Lemma 1.5.5 ([67, Lemma 1]). Let Y ⊆ Cn be a constructible subset of codimension e. LetF1, . . . , Fp, g1, . . . , gp : Cn 99K C be rational functions such that g1, . . . , gp never vanish at anypoint of Y . Then for generic (c1, . . . , cp) ∈ Kp,

codim(Y ∩ V (F1 + c1g1, . . . , Fp + cpgp)) > e+ p.

In particular, if p > n, then Y ∩ V (F1 + c1g1, . . . , Fp + cpgp) = ∅ for generic (c1, . . . , cp).

Let us consider F = (F1, . . . , Fn) : Cn → Cn be a polynomial map and h ∈ C[x1, . . . , xn].Then for all α = (α1, . . . , αn) ∈ Cn, we denote by F + hα the polynomial map Cn → Cn

given by F + hα = (F1 + α1h, . . . , Fn + αnh).

Theorem 1.5.6. Let F = (F1, . . . , Fn) : Cn → Cn be a finite polynomial map and leth ∈ C[x1, . . . , xn]. Then there exists an open Zariski-dense subset U ⊆ Cn such that F +hα :Cn → Cn is a finite polynomial map, for all α = (α1, . . . , αn) ∈ U .

Proof. Let us consider the Zariski-open set Y = Cn \ V (F1) ∪ · · ·V (Fn) ∪ V (h). Then, byLemma 1.5.5 we have there exists a dense subset U in Cn, such that Y ∩V (F1 +α1h, . . . , Fn+αnh) is a finite set for all (α1, . . . , αn) ∈ U . Also, it is immediate to check that

(Cn \ Y ) ∩ V (F1 + α1h, . . . , Fn + αnh) ⊆ V (F1, . . . , Fn),

for all α = (α1, . . . , αn) ∈ U ∩ (C∗)n. Since V (F1, . . . , Fn) is a finite set, the result followstaken V = U ∩ (C∗)n, which is clearly dense.


1.6 Łojasiewicz exponent at infinity of polynomial maps

From now on, we denote K = R or C. In this section we expose some basic definitionsand results concerning to Łojasiewicz exponents at infinity of a polynomial map Kn → Kp.For a detailed account on this subject we refer to the survey article [44]).

Definition 1.6.1. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. Let us considerthe set e∞(F ) formed by those ν ∈ R such that there exist positive constants B and C suchthat

‖F (z)‖ > C‖z‖ν , (1.12)

for all z ∈ Kn such that ‖z‖ > B. If e∞(F ) 6= ∅, we denote by L∞(F ) the supremum ofe∞(F ). We define L∞(F ) = −∞ when e∞(F ) = ∅. The number L∞(F ), when finite, is calledthe Łojasiewicz exponent at infinity of F .

The next proposition gives a basic property that characterizes the finiteness of the Ło-jasiewicz exponent at infinity.

Proposition 1.6.2 ([44, Proposition 2.1]). Let F = (F1, . . . , Fp) : Cn → Cp be a polynomialmap. Then L∞(F ) 6= −∞ if and only if the set F−1(0) is compact.

The above proposition is also valid for real polynomial maps and a proof of this fact canbe found in [8, p. 749]. By this result we assume from now on that F−1(0) is a compact set.

Example 1.6.3. The following examples can be found in [44] and [60].

(i) For F (x, y) = (x, xy − 1) : C2 → C2, then we have L∞(F ) = −1.

(ii) Take p, q, s ∈ N, such that s > qp. Let F : C2 → C2 be a polynomial map, given by

F (x, y) = (xs, xps + yq). ThenL∞(F ) = q

p

(iii) Take p, q ∈ N, 0 < q < p. Let g be the polynomial of C[x, y] given by g(x, y) =y + y1+qxp−q. Let us consider the following polynomial map G given by G(x, y) =grad g = ((p− q)y1+qxp−q−1, 1 + (1 + q)yqxp−q) : C2 → C2. Then

L∞(G) = −pq.

In general, it is well known that the Łojasiewicz exponent at infinity of a polynomial mapis a rational number, as we show below.


Definition 1.6.4. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map and S ⊆ Kn be anunbounded set. Let us consider the set eS,∞(F ) formed by those ν ∈ R such that there existpositive constants R and C such that

‖F (z)‖ > C‖z‖ν , (1.13)

for all z ∈ S such that ‖z‖ > B. If eS,∞(F ) 6= ∅, we denote by L∞(F |S) the supremum ofeS,∞(F ). We define L∞(F |S) = −∞ when eS,∞(F ) = ∅. The number L∞(F |S), when finite,is called The Łojasiewicz exponent at infinity on S of F

Example 1.6.5. For F (x1, x2) = (x1, x1x2 − 1) : C2 → C2 we have L∞(F ) = −1. IfS := (x1, x2) ∈ C2 : x2 = 0, then L∞(F |S) = 0.

By a meromorphic curve in Kn we mean a set Γ ⊆ Kn for which there exists a holomorphicmapping (called a parametrization of Γ)

Φ = (ϕ1, . . . , ϕp) : t ∈ K : |t| > R → Kn,

for some R > 0, such that ImΦ = Γ and Φ has a pole at ∞ ∈ K (it means that each ϕi hasat most a pole at ∞ and at least one of ϕi has a pole at ∞, this implies that ‖Φ(t)‖ → ∞when t→∞). Since each ϕi 6= 0 has an expansion in a Laurent series at ∞, we can write ϕias

ϕi(t) =k0∑

k=−∞akt

k, ak0 6= 0, k0 ∈ Z.

Then, we define degϕi := k0 and deg Φ := maxdegϕ1, . . . , degϕp.The next result shows the connection between the above definitions.

Proposition 1.6.6 ([44, Proposition 2.5]). Let F : Kn → Kp be a polynomial map. ThenL∞(F ) is attained on a meromorphic curve at infinity i.e.

L∞(F ) = L∞(F |Γ),

where Γ is a meromorphic curve at infinity.

If Γ is a meromorphic curve at infinity on which L∞(F ) is attained, then it is easy toobserve that

L∞(F |Γ) = deg(F Φ)deg Φ . (1.14)

Thus as a consequence of the above proposition and from the last equation we have thefollowing result.


Proposition 1.6.7. Let F : Kn → Kp be a polynomial map. Then L∞(F ) is a rationalnumber.

Proof. It follows immediately from the above Proposition 1.6.6 and the equality (1.14) takingan appropriate meromorphic curve at infinity.

Proposition 1.6.8 ([44]). Let F : Kn → Kp be a polynomial map. Then L∞(F ) is attainedi.e. for the number L∞(F ), there exist positive constants B and C such that

‖F (z)‖ > C‖z‖L∞(F ),

for all z ∈ Kn, such that ‖z‖ > B.

Proposition 1.6.9 ([44]). Let F : Cn → Cp be polynomial map. Then L∞(F ) is an invariantof linear (not affine) automorphisms of Cn and Cp.

Remark 1.6.10. In the above proposition, L∞(F ) is not an invariant of polynomial auto-morphisms of the domain Cn and the codomain Cp, as we show in following examples.

(i) Triangular automorphisms of Cn may change L∞(F ). For example L∞(x1, x2) = 1 andL∞(x1, x2 + x2

1) = 12 .

(ii) Translations in Cp may change L∞(F ). For example L∞(x1, x1x2 − 1) = −1 andL∞(x1, x1x2) = −∞.

Proposition 1.6.11 ([44]). Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. ThenL∞(F ) 6 degF , where degF := maxdegF1, . . . , degFp.

The above proposition follows from the elementary inequality for polynomial maps, i.e.there exist positive constants B and C such that

‖F (z)‖ 6 C‖z‖degF ,

for all z ∈ Kn, such that ‖z‖ > B.The most complete results about the computation of the Łojasiewicz exponent at infinity

of a polynomial map F : Cn → Cp appear in the case n = p = 2 (the case n = 1 is trivial sincein this case L∞(F ) = degF ). Formulas for L∞(F ) were given in various terms. The first onewas a formula for L∞(F ) in terms of parametrizations of the branches of the zero-sets of F1

and F2 at infinity.


Theorem 1.6.12 ([23]). Let F : C2 → C2 be a polynomial map. If F is not constant, then

L∞(F ) = L∞(F |V (F1) ∪ V (F2))

where V (F1) = F−11 (0) and V (F2) = F−1

2 (0) are zero-sets of F1 and F2.

To reformulate this theorem in terms of parametrizations, we notice that V (F1) and V (F2)are complex algebraic curves in C2 (provided that F1 and F2 are not constants). Then, in aneighborhood of infinity in C2 they are finite unions of meromorphic curves (called branchesof F1 and F2 at infinity, see [23, Proposition 3.1]. If we denote by Φi, i = 1, . . . , r (resp.Ψj, j = 1, . . . , s) parametrizations of the branches of F1 (resp. F2) at infinity then we havethe following equivalent form for the previous theorem.

Theorem 1.6.13. Let F : C2 → C2 be a polynomial map. If F is not constant, then

L∞(F ) = mini,j

degF Φi

deg Φi

,degF Ψj

deg Ψj

Let us pass to the n-dimensional case. So far, there are no explicit formulas for theŁojasiewicz exponent at infinity of F = (F1, . . . , Fp) : Cn → Cp. But we give a generalizationof Theorem 1.6.12.

Theorem 1.6.14 ([24, Theorem 1]). Let F = (F1, . . . , Fp) : Cn → Cp be a polynomial map.If F is not constant, then

L∞(F ) = L∞(F |V (F1) ∪ · · · ∪ V (Fp)).

In many cases one needs only estimations of the Łojasiewicz exponent at infinity in termsof the other numerical invariants related to F .

In general when F = (F1, . . . , Fp) : Cn → Cp is a polynomial map, with n 6= p, then wehave a lower estimate for L∞(F ), as we describe below.

Definition 1.6.15. Let F = (F1, . . . , Fp) : Cn → Cp be a polynomial map and di := degFifor i = 1, . . . p. Then we define

B(d1, . . . , dp, n) =d1 · · · dp−1 · dp if p 6 nd1 · · · dn−1 · dp if p > n.

Many papers contain estimates of L∞(F ) for various classes of polynomial map [45], [60],[61]. For instance the most interesting estimate was obtained by Kollár in [45]( Theorem1.10, proves this but it is stated incorrectly see [69, Remark 18]), show the following.


Theorem 1.6.16. Let F = (F1, . . . , Fp) : Cn → Cp be a polynomial map such that F−1(0) isa finite set and degFi = di > 0, for all i = 1, . . . , p. Let us suppose that d1 > · · · > dp. Then

L∞(F ) > dp −B(d1, . . . , dp, n)

Cygan Ewa, Krasiński and Tworzewski in [25] give a direct generalization of the aboveresult.

Theorem 1.6.17 ([25, Theorem 7.3]). In the same conditions of Theorem 1.6.16, we havethat

L∞(F ) > dp −B(d1, . . . , dp, n) +∑

b∈F−1(0)µb(F )

where µb(F ) denotes the multiplicity of F at an isolated zero b.

In the above results lower bounds for the Łojasiewicz exponent at infinity are given in termsof the degree of the component functions and the multiplicity of the polynomial map.

Next, we present some applications of the Łojasiewicz exponent at infinity for polynomialmaps and in particular we introduce the Jacobian Conjecture.

Proposition 1.6.18 ([44, Proposition 4.1]). Let F = (F1, . . . , Fp) : Cn → Cp be a polynomialmap. Then F is a proper mapping if and only if L∞(F ) > 0.

The above proposition characterizes the properness of polynomial maps in terms of itsŁojasiewicz exponent. Given a map F : Kn → Kp, we recall that F is said to be proper ifand only if for any compact set S ⊆ Kp the preimage F−1(S) ⊆ Kn is also compact (it isequivalent to the property that for any sequence zkk∈N ⊆ Kn such that ‖zk‖ → ∞ we have‖F (zk)‖ → ∞). This result is also valid for real polynomial maps and the proof in this caseis completely analogous to the complex case. Let us remark that the if part is immediateand the converse follows as a consequence of the Curve Selection Lemma at infinity (see [57,Lemma 2] and [44] for details).

A motivation to study the Łojasiewicz exponent at infinity is the famous “Jacobian Con-jecture”. This conjecture states that if F = (F1, . . . , Fn) : Cn → Cn is a polynomial mapand det(DF ) is a not-zero constant in Cn, then F is a polynomial automorphism, where DFdenotes the differential matrix of F . This conjecture has been approached in different areas ofmathematics, as can be seen in the monograph of Van Den Essen [76]. We recall that Pinchukshowed an example of polynomial map F : R2 → R2 such that F is a local homeomorphismat each point (x, y) ∈ R2, but F is not injective [59].


The Jacobian conjecture can be reduced to the problem on the Łojasiewicz exponent bythe following proposition.

Proposition 1.6.19 ([44, Proposition 4.2]). Let F = (F1, . . . , Fn) : Cn → Cn be a polynomialmap such that det(DF ) is a not-zero constant in Cn, then F is a polynomial automorphismof Cn if and only if L∞(F ) > 0.

Hence to prove Jacobian Conjecture it is enough to check if F is proper. By the aboveproposition if F is a polynomial automorphism of Cn then L∞(F ) > 0. In this case, we canask, what positive rational number is it? The answer was given by Płoski in [60].

Proposition 1.6.20 ([60]). Let F : Cn → Cn be a polynomial automorphism of Cn, then

L∞(F ) = 1degF−1

CHAPTER 2

Newton filtrations and non-degeneracyconditions

Contents2.1 Global Newton polyhedra . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Newton non-degeneracy at infinity . . . . . . . . . . . . . . . . . 32

2.3 The Newton filtration . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Counting affine roots of polynomial systems . . . . . . . . . . . . 40

The fundamental concept that we study in this chapter is the Newton non-degeneracy atinfinity of polynomial maps F : Kn → Kp. As we will see, we generalize this notion in viewof several purposes. In this section we also introduce the notion of Newton non-degeneracywith respect to a fixed convenient Newton polyhedron at infinity. The motivation for thematerial of this section have been the articles [11], [43] and [67].

2.1 Global Newton polyhedra

In this section we describe some combinatorial concepts about Newton polyhedra at infinity.

Definition 2.1.1. A subset Γ+ ⊆ Rn>0 is said to be a global Newton polyhedron or a Newton

polyhedron at infinity, if there exists some finite subset A ⊆ Zn>0 such that Γ+ is equal to theconvex hull in Rn of A ∪ 0.

25

26 Chapter 2. Newton filtrations and non-degeneracy conditions

Let us fix a compact set S ⊆ Rn. If w ∈ Rn, then we define

`(w, S) = min〈w, k〉 : k ∈ S

, (2.1)

∆(w, S) =k ∈ S : 〈w, k〉 = `(w, S)

. (2.2)

where we denote by 〈 , 〉 the standard scalar product in Rn. If w ∈ Rn \ 0, then ∆(w, S) iscalled the face of S supported by w. The hyperplane given by the equation 〈w, k〉 = `(w, S)is called a supporting hyperplane of S.

Let Γ+ ⊆ Rn>0 be a global Newton polyhedron and let ∆ be a face of Γ+. The dimension

of the face ∆, denoted by dim(∆), is defined as the minimum among the dimensions of theaffine subspaces containing ∆. We denote by Γ the union of the faces of Γ+ not containingthe origin. We remark that Γ is not a convex set in general. The faces of Γ+ of dimension 0are called vertices of Γ+ and the faces of Γ+ of dimension n− 1 are called facets of Γ+.

We define the dimension of Γ as

dim(Γ) = max

dim(∆) : ∆ is a face of Γ+ such that 0 /∈ ∆.

Example 2.1.2. Let e1, e2, e3 be the canonical basis in R3, let c = e1 + e2 + e3 and let usfix an integer a > 1. Let us consider the following subsets of Z3

>0:

A1 = 2ae1, 2ae2, 2ae3, 2a(1, 1, 0), 2a(1, 0, 1), 2a(0, 1, 1), a(2, 1, 2), a(2, 2, 1), a(1, 2, 2).

A2 = 3ae3, ac, a(1, 1, 0).

Let Γ1+ = Γ+(A1) and let Γ2

+ = Γ+(A2). Figure (2.1) shows a picture of Γ1+ and Γ2

+. Weremark that dim Γ1

+ = 2 and dim Γ2+ = 1.

o

Γ+~2

(0,0,3a)

e3e3

-e1

e2-e2

e1

3

0

Γ~

+1

2a,0,0 2a,2a,0

2a,2a,a

2a,a,2a

0,0,2a 0, 2a, 2a

-c

-e

-1,-1,0

c

x1

x2

3xx3

x1

x2

( )

a 2a 2a( , , )

( )

( )

( )

2a,0,2a( )

( )

a,a,0( )

a,a,a( )( )

( )

Figure 2.1: Newton polyhedra at infinity

Chapter 2. Newton filtrations and non-degeneracy conditions 27

For any w = (w1, . . . , wn) ∈ Rn, we denote by w0 the minimum of the coordinates of w.Then we define Rn

0 = w ∈ Rn : w0 < 0 and Rn0 (i) = w ∈ Rn

0 : w0 = wi, for all i = 1, . . . , n.Let us remark that if Γ+ ⊆ Rn is a global Newton polyhedron then

Γ+ = k ∈ Rn>0 : 〈k, w〉 > `(w, Γ+), for all w ∈ Rn

0. (2.3)

Let w ∈ Zn. We say that w is primitive when w 6= 0 and w is the vector of smallest lengthbetween all vectors of Zn of the form λw, for some λ > 0.

Let Γ+ be a Newton polyhedron at infinity in Rn such that dim(Γ) = n − 1. We denoteby F(Γ+) the family of primitive vectors w ∈ Zn such that dim ∆(w, Γ+) = n− 1. Since Γ+

is the convex hull of a finite subset of Rn and dim(Γ) = n − 1 then F(Γ+) is a finite familyof vectors (see [31, p. 29]). Moreover, any face of Γ+ can be expressed as an intersection∩w∈J∆(w, Γ+), for some subset J ⊆ F(Γ+) (see [31, p. 33]). We denote by F0(Γ+) the subsetof F(Γ+) formed by the vectors w ∈ F(Γ+) such that ∆(w, Γ+) does not contain the origin.

Lemma 2.1.3. Let Γ+ ⊆ Rn>0 be a global Newton polyhedron. Let J be a subset of F(Γ+).

Then the following conditions are equivalent:

(i) ∩w∈J∆(w, Γ+) 6= ∅;

(ii) ∩w∈J∆(w, Γ+) = ∆(∑

w∈J w, Γ+);

(iii) `(∑w∈J w, Γ+) = ∑w∈J `(w, Γ+).

Proof. The result follows as a direct consequence of the definition of `(w, Γ+) and ∆(w, Γ+),for a given vector w ∈ Rn.

Let Γ1+, . . . , Γ

p+ be global Newton polyhedra in Rn. Then theMinkowski sum of Γ1

+, . . . , Γp+

is defined as Γ1+ + · · · + Γp+ = k1 + · · · + kp : ki ∈ Γi+, for all i = 1, . . . , p. It is well known

that Γ1+ + · · ·+ Γp+ is again a global Newton polyhedron. The definition of A1 + · · ·+Ap for

arbitrary sets A1, . . . , Ap ⊆ Rn is analogous.

Lemma 2.1.4. Let Γ1+, . . . , Γ

p+ be global Newton polyhedra in Rn. Let Γ+ = Γ1

+ + · · · + Γp+and let w ∈ Rn \ 0. Then

(i) `(w, Γ+) = `(w, Γ1+) + · · ·+ `(w, Γp+);

(ii) ∆(w, Γ+) = ∆(w, Γ1+) + · · ·+ ∆(w, Γp+).

Proof. It arises as a consequence of the definition of Minkowski sum.


Let e1, . . . , en denote the canonical basis in Rn. Given a global Newton polyhedronΓ+ ⊆ Rn

>0, we say that Γ+ is convenient if Γ+ intersects each coordinate axis in a pointdifferent from the origin, that is, if for any i ∈ 1, . . . , n there exists some α > 0 such thatαei ∈ Γ+. In this case we define

ri(Γ+) = maxr > 0 : rei ∈ Γ+, i = 1, . . . , n. (2.4)

r0(Γ+) = minr1(Γ+), . . . , rn(Γ+)

. (2.5)

Lemma 2.1.5. Let Γ+ be a convenient global Newton polyhedron in Rn. Let w ∈ Rn. Thenthe following conditions are equivalent:

(i) 0 /∈ ∆(w, Γ+);

(ii) `(w, Γ+) < 0;

(iii) w0 < 0.

Proof. It follows easily using the fact that Γ+ is convenient (see [8, Lemma 4.2]).

Lemma 2.1.6. Let Γ+ be a convenient Newton polyhedron at infinity, Γ+ ⊆ Rn>0. Then

ri(Γ+) = min`(w, Γ+)

wi: w ∈ Rn

0 (i), for all i = 1, . . . , n. (2.6)

Proof. Equality (2.6) follows as an immediate consequence of Lemma 2.1.5 and the fact thatΓ+ is expressed as Γ+ = k ∈ Rn

>0 : 〈w, k〉 > `(w, Γ+), for all w ∈ Rn0.

Let us fix coordinates x1, . . . , xn in Kn and let k ∈ Zn>0. Then we write xk to denote themonomial xk1

1 · · · xknn .

Definition 2.1.7. Let h ∈ K[x1, . . . , xn], h 6= 0. Let us suppose that h is written as h =∑k akx

k. Then the support of h, denoted by supp(h), is defined as the set of those k ∈ Zn>0

such that ak 6= 0. If w ∈ Rn, then

`(w, h) = `(w, supp(h)), (2.7)

∆(w, h) = ∆(w, supp(h)). (2.8)

Let c = (1, . . . , 1) ∈ Rn>0, then we observe that `(−c, h) = − deg(h). The Newton polyhedron

at infinity of h is defined as the convex hull of supp(h) ∪ 0 and is denoted by Γ+(h). Wesay that h is convenient when Γ+(h) is convenient. If h = 0, then we set supp(h) = ∅ andΓ+(h) = ∅.


If we consider a map F = (F1, . . . , Fp) : Kn → Kp, then the Newton polyhedron at infinityof F , that we denote by Γ+(F ), is defined as the convex hull of Γ+(F1) ∪ · · · ∪ Γ+(Fp). Wesay that F is convenient when Γ+(F ) is convenient. We denote by Γ(F ) the union of thosefaces of Γ+(F ) not passing through the origin.

Next we recall some important concepts from convex geometry.

Definition 2.1.8. If A is a subset of Rn, then we denote by Conv(A) the convex hull of A inRn. If P ⊆ Rn, then we say that P is a polytope when there exists a finite subset A ⊆ Rn suchthat P = Conv(A). If A is contained in Zn, then Conv(A) is said to be a lattice polytope. Inparticular, we observe that any global Newton polyhedron in Rn is a lattice polytope. If P isa polytope in Rn, then the dimension of P is defined as the minimum dimension of an affinesubspace containing P .

If K ⊆ Rn is a compact set, then we denote by Vn(K) the n-dimensional volume ofK. Let C1, . . . , Cn be polytopes in Rn and let λ1, . . . , λn ∈ R>0. Let λ1C1 + · · · + λnCn =λ1k1 + · · ·+ λnkn : ki ∈ Ci, i = 1, . . . , n. It is a classical result from convex geometry thatVn(λ1C1 +· · ·+λnCn) is a homogeneous polynomial of degree n in the variables λ1, . . . , λn(seefor instance [20, p. 337]). The n-dimensional mixed volume of C1, . . . , Cn is defined as thecoefficient of λ1 · · ·λn in the polynomial Vn(λ1C1 + · · · + λnCn). We denote this number byMV(C1, . . . , Cn). Let us recall some elementary properties of this number

(i) MV(C1, . . . , Cn) is symmetric and linear in each variable.

(ii) MV(C1, . . . , Cn) > 0 and MV(C1, . . . , Cn) = 0 if and only if dim(∑i∈I Ci) < |I| for somenonempty subset I ⊆ 1, . . . , n, where |I| denotes the cardinal of I.

(iii) MV(C1, . . . , Cn) ∈ Z>0, if Ci is a lattice of polytope, for all i = 1, . . . , n.

(iv) MV(C, . . . , C) = n!Vn(C) for any polytope C ⊆ Rn.

If P is a polytope in Rn, then we denote Conv(P ∪ 0) by P 0. If P = (P1, . . . , Pn) is ann-tuple of polytopes of Rn, then we define P0 = (P 0

1 , . . . , P0n). In particular, it makes sense to

speak about the mixed volumes MV(P) and MV(P0). In general MV(P) 6 MV(P0), by themonotonicity of mixed volumes with respect to inclusion. We refer to [21, 31, 65] for moreinformation about MV(C1, . . . , Cn).


2.2 Newton non-degeneracy at infinity

Let Γ+ be a global Newton polyhedron of Rn and let ∆ a face of Γ+. If h ∈ K[x1, . . . , xn]and h is written as h = ∑

k akxk, then we denote by h∆ the polynomial obtained as the sum

of those terms akxk such that k ∈ ∆. If supp(h) ∩∆ = ∅, then we set h∆ = 0.Motivated by the concept of Newton non-degenerate polynomial function [43, p. 7] we

introduce the following definition.

Definition 2.2.1. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. We say that F isNewton non-degenerate at infinity when

x ∈ Kn : (F1)∆(x) = · · · = (Fp)∆(x) = 0 ⊆ x ∈ Kn : x1 · · ·xn = 0, (2.9)

for all compact face ∆ of Γ+(F ) not containing the origin.

If F : Kn → Kp is a polynomial map, then we denote by I(F ) the ideal of K[x1, . . . , xn]generated by the component functions of F .

Example 2.2.2. Let us consider the polynomial maps F1, F2 : C2 → C2 given by F1(x1, x2) =(x1, x

21 + x2

2) and F2(x1, x2) = (x1, x22), for all (x1, x2) ∈ C2. It is immediate to see that I(F1)

and I(F2) are generating the same ideal in C[x1, x2]. However, F1 is Newton degenerate atinfinity and F2 is Newton non-degenerate at infinity.

The above easy example shows that the Newton non-degeneracy at infinity can not beextended to ideals of K[x1, . . . , xn] by imposing this condition to a fixed generating system ofa given ideal, which is not the case when dealing with the analogous concept in the ring ofanalytic germs (Kn, 0)→ K.

Unless otherwise stated, in the remaining section we will consider only the case K = C.Let us fix a global Newton polyhedron Γ+ ⊆ Rn. If ∆ is a face of Γ+, then we denote byC(∆) the cone over ∆, that is, the the union of all half lines emanating from the origin andpassing through some point of ∆.

Suppose that ∆ does not contain the origin. Then, we denote by R∆ the subring ofC[x1, . . . , xn] given by all h ∈ C[x1, . . . , xn] such that supp(h) ⊆ C(∆). We remark that R∆

is a Cohen-Macaulay ring of dimension dim(∆) + 1 (see [43, p. 24]).

Proposition 2.2.3 ([43, Théorème 6.2]). Let us fix a convenient Newton polyhedron Γ+ ⊆Rn

>0. Let F = (F1, . . . , Fp) : Cn → Cp be a polynomial map. If ∆ is a face of Γ+ notcontaining the origin, then the following conditions are equivalent:


(i) The ideal of R∆ generated by (F1)∆, . . . , (Fp)∆ has finite colength.

(ii) For each face ∆1 of Γ+ such that ∆1 ⊆ ∆, we have

x ∈ Cn : (F1)∆1(x) = · · · = (Fp)∆1(x) = 0 ⊆ x ∈ Cn : x1 · · ·xn = 0.

The next result is an immediate consequence of Definition 2.2.1 and Proposition 2.2.3 andshows an alternative formulation of the Newton non-degeneracy at infinity of complex mapsF : Cn → Cp.

Corollary 2.2.4. Let F : Cn → Cp be a polynomial map. Then F is Newton non-degenerateat infinity if and only if for each face ∆ of Γ+ not containing the origin, the ideal of R∆

generated by (F1)∆, . . . , (Fp)∆ has a finite colength.

If F : Cn → Cp is Newton non-degenerate at infinity, then Corollary 2.2.4 allows us togive an estimation of p.

Corollary 2.2.5. Let F : Cn → Cp is a polynomial map such that F is Newton non-degenerate at infinity. Then p > dim(Γ(F )) + 1.

Proof. Let us consider ∆ a face of Γ+ such that 0 /∈ ∆ and dim Γ(F ) = dim(∆). Thendim(R∆) = dim(∆) + 1 (see [15, p. 210]). Since F is Newton non-degenerate at infinity,by Corollary 2.2.4, the ideal I generated by (F1)∆, . . . , (Fp)∆ has a finite colength in R∆,which implies that I must be generated at least dim(∆) + 1 non-zero elements of R∆, by [53,p. 77]. Therefore p > dim(∆) + 1 = dim(Γ) + 1.

As a consequence of the previous result, if F : Cn → Cp is a convenient polynomial mapsuch that F is Newton non-degenerate at infinity, then p > n, since dim(Γ(F )) = n − 1 inthis case.

Remark 2.2.6. It is immediate to see that the above corollary is not true when K = R. Forthis, let us consider the polynomial map F : R2 → R, given by F (x1, x2) = x2

1 + x22. Then it

is easy to check that F is Newton non-degenerate at infinity and dim(Γ(F )) = 1.

Definition 2.2.7. Let f ∈ K[x1, . . . , xn]. Let us denote by A(f) the polynomial map Kn →Kn given by

A(f) =(x1∂f

∂x1, . . . , xn

∂f

∂xn

).

The function f is said to be Newton non-degenerate at infinity when the map A(f) is Newtonnon-degenerate at infinity.


Let us remark that Γ+(f) = Γ+(A(f)), for all f ∈ K[x1, . . . , xn]. Then above definitioncoincides with the notion of non-degeneracy established for polynomial functions introducedby Kouchnirenko in [43, p. 7].

Let Γ+ ⊆ Rn>0 be a convenient Newton polyhedron at infinity and let C[Γ+] be denote

the set of polynomial functions Cn → C whose support is contained in Γ+. Then it is well-known, by [43, Théorème 6.1], that the class of polynomial maps of C[Γ+] which are Newtonnon-degenerate at infinity constitute a dense Zariski-open set of C[Γ+].

Let Γ+ ⊆ Rn>0 be a convenient Newton polyhedron at infinity. Then the Newton number

of Γ+ is defined, in [43, p. 4], as

ν(Γ+) = n!Vn − (n− 1)!Vn−1 + · · ·+ (−1)n−11!V1 + (−1)n,

where Vi denotes the sum of the i-dimensional volumes of the intersections of Γ+ with thecoordinates planes of dimension i, for i = 1, . . . , n. If f ∈ C[x1, . . . , xn] has a finite numberof singular points, then the Milnor number of f at infinity is defined as

µ∞(f) = dimCC[x1, . . . , xn]

J(f) ,

where J(f) is the ideal in C[x1, . . . , xn] generated by the derivatives ∂f∂x1, . . . , ∂f

∂xn.

Here we state a known result of Kouchnirenko about the computation of the Milnornumber of a Newton non-degenerate polynomial function at infinity in terms of Γ+(f).

Theorem 2.2.8 ([43]). Let f : Cn → C be a convenient polynomial function with a finitenumber of singular points. Then µ∞(f) 6 ν(Γ+(f)), and equality holds if f is Newton non-degenerate at infinity.

The explicit expresion of µ∞(f) in terms of Γ+(f) for non convenient polynomial functionsCn → C is a hard problem. This task has been done by P. Cassou-Noguès [17] and Artal-Luengo-Melle [1].

The local counterpart of the above result was the motivation of Bivià-Ausina, Fukui andSaia in [11] to analyze under which conditions the multiplicity of (non-gradient) analytic mapgerms (Cn, 0) → (Cn, 0), or more generally, the codimension of determinantal ideals, couldbe computed by means of a given (local) Newton filtration. This kind of results help inthe study and the effective computation of numerical invariants of singularities of complexanalytic maps (Cn, 0)→ (Cp, 0).


The main goal of this chapter is to extend Theorems 3.3 and 3.6 of [11] to the context ofpolynomial maps Cn → Cn. For this purpose we introduce in the next section the notion offiltration of C[x1, . . . , xn] associated to a convenient Newton polyhedron in Rn

>0.

2.3 The Newton filtration

Along this section, we denote by Γ+ a fixed convenient Newton polyhedron at infinitycontained in Rn

>0. Let ∆ denote a face of Γ+ not containing the origin. Then, as in theprevious chapter, we denote by C(∆) the cone over ∆, that is, the union of all half linesemanating from the origin and passing through some point of ∆.

It is immediate to see that F(Γ+) = F0(Γ+) ∪ e1, . . . , en. Let F0(Γ+) = w1, . . . , wr,for some wi ∈ Zn ∩ Rn

0 , i = 1, . . . , r, and let MΓ+denote the least common multiple of

−`(wi, Γ+) : i = 1, . . . , r. We remark that `(wi, Γ+) < 0, for all i = 1, . . . , r (see Lemma2.1.5). Then, following Kouchnirenko in [43], we define the linear map φi : Rn

>0 → R, by

φi(k) =MΓ+

`(wi, Γ+)〈wi, k〉,

for all k ∈ Rn>0 and all i = 1, . . . , r. Thus, we define the map φΓ+

: Rn>0 → R by

φΓ+(k) = max

16i6rφi(k) = MΓ+

max16i6r

〈wi, k〉`(wi, Γ+)

,

for all k ∈ Rn>0. We will refer to φΓ+

as the filtrating map associated to Γ+. If no confusionarises, we denote MΓ+

and φΓ+simply by M and φ, respectively. We have that φ satisfies

the following properties:(i) φ(Zn>0) ⊆ Z>0,

(ii) φ is constant on the boundary Γ and equal to M .

Lemma 2.3.1. Under the above conditions, let k ∈ Rn>0 and let i0 ∈ 1, . . . , r. Then

φ(k) = φi0(k) if and only if k ∈ C(∆(wi0 , Γ+)).

Proof. Since Γ+ is convenient, we can write k as k = λk′, for some k′ ∈ Γ. A simplecomputation shows that φ(k) = φi0(k) if and only if

〈k′, wi〉`(wi, Γ+)

6〈k′, wi0〉`(wi0 , Γ+)

,

for all i ∈ 1, . . . , r. But this last condition is equivalent to saying that 〈k′,wi0 〉`(wi0 ,Γ+)

= 1, which,in particular says that k′ ∈ ∆(wi0 , Γ+), and then the result follows.


The above lemma shows that φ is linear on each cone C(∆), where ∆ is any face of Γ+

not passing through the origin. Moreover it is immediate to see that φ(a+ b) 6 φ(a) + φ(b),for all a, b ∈ Rn

>0, and, by Lemma 2.3.1, equality holds if and only if a and b belong to thesame cone, that is, there exists a vector w ∈ F0(Γ+) such that a, b ∈ C(∆(w, Γ+)).

Let us remark that

Γ+ =k ∈ Rn

>0 : 〈wi, k〉 > `(wi, Γ+), for all i = 1, . . . , r

= k ∈ Rn>0 : φ(k) 6M.

Definition 2.3.2. We define the Newton filtration of C[x1, . . . , xn] associated to Γ+ as themap νΓ+

: C[x1, . . . , xn] → Z+ ∪ −∞ given by νΓ+(h) = maxφΓ+

(k) : k ∈ supp (h)when h 6= 0 and νΓ+

(0) = −∞. The number νΓ+(h) is called the degree of h with respect

to Γ+. Let us remark that when Γ+ is equal to the standard n-simplex, that is, whenΓ+ = Γ+(e1, . . . , en), then νΓ+

(h) coincides with the usual notion of degree of h, for allh ∈ C[x1, . . . , xn]. If no confusion arises, we denote the map νΓ+

simply by ν.

We observe that the map ν : C[x1, . . . , xn]→ Z+∪−∞ satisfies the following properties:

(i) ν(1) = 0, ν(0) = −∞;

(ii) ν(h+ g) 6 maxν(h), ν(g);

(iii) ν(h · g) 6 ν(h) + ν(g).

The above properties are straightforward consequences of the properties of the filtrating mapφ. Let us define, for all r > 0, the following set of polynomials:

Ar = h ∈ C[x1, . . . , xn] : ν(h) 6 r ∪ 0 (2.10)

and we set Ar = 0 for all r < 0. We denote by Γ+(Ar) the convex hull of k ∈ Zn>0 : φ(k) 6r ∪ 0. In particular A0 = C. By the properties of φ it is immediate to check that thecollection Arr∈Z verify

(i) Ar is an additive subgroup of C[x1, . . . , xn], for all r > 0;

(ii) Ar ⊆ Ar+1, for all r > 0;

(iii) ArAs ⊆ Ar+s, for all r, s ∈ Z;

(iv) Γ+(Ar) ⊆ rM

Γ+ and equality holds if and only if Vn(Γ+(Ar)) = ( rM

)nVn(Γ+), where Vn

denotes the n-dimensional volume.


The collection Arr∈Z is also known as the filtration of C[x1, . . . , xn] associated to Γ+.From this filtration we can construct the graded ring A = ⊕r>0Ar, where Ar = Ar/Ar−1,for all r > 0.

The objective of this section is to obtain a characterization of an important class ofpolynomial maps Cn → Cn that extends the class of pre-weighted homogeneous maps (seeDefinition 2.3.5). In particular, we obtain a version of [11, Theorem 3.3] in the ring of poly-nomials C[x1, . . . , xn]. In the remaining section Γ+ denotes a fixed global Newton polyhedronin Rn. Let us denote by ν the filtration νΓ+

: C[x1, . . . , xn]→ Z>0 ∪ −∞.

Definition 2.3.3. Let h ∈ K[x1, . . . , xn], h 6= 0. Let us suppose that h is written as h =∑k akx

k. Let ∆ be a face of Γ+ not passing through the origin. Then the principal part ofh over ∆, denoted by in∆(h), is the polynomial obtained as the sum of all terms akxk suchthat k ∈ C(∆) and ν(h) = ν(xk). If no such terms exist or h = 0, then we set in∆(h) = 0.We observe that if h∆ 6= 0, then h∆ = in∆(h) if and only if νΓ+

(h) = MΓ+

Motivated by Bivià-Ausina, Fukui and Saia [11], we introduce the following concept.

Definition 2.3.4. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. We say that F isnon-degenerate with respect to Γ+ when

x ∈ Cn : in∆(F1)(x) = · · · = in∆(Fp)(x) = 0 ⊆ x ∈ Cn : x1 · · ·xn = 0, (2.11)

for all faces ∆ of Γ+ not containing the origin.

We remark that the previous definition, which is specially significant when p = n, consti-tutes a generalization of the notion of pre-weighted homogeneous map from Cn to Cp.

Definition 2.3.5. Let us fix a primitive vector v = (v1, . . . , vn) ∈ Zn>1, v 6= 0 and leth ∈ K[x1, . . . , xn]. Let us suppose that h is written as h = ∑

k akxk. Let F = (F1, . . . , Fp) :

Kn → Kp be a polynomial map.

(i) Let us denote the integer max〈v, k〉 : k ∈ supp(h) by dv(h). We call dv(h) the degreeof h with respect to v. We define the principal part of h at infinity with respect to v,denoted by qv(h) as the sum of those terms akxk such that 〈v, k〉 = dv(h). If h = 0,then we set dv(h) = −∞ and qv(h) = 0. We define qv(F ) = (qv(F1), . . . , qv(Fp)) anddv(F ) = (dv(F1), . . . , dv(Fp)).

(ii) If v ∈ Zn>1. We say that h is weighted-homogeneous of degree d with respect to v whensupp(h) is contained in the hyperplane of equation 〈v, k〉 = d. That is, when qv(h) = h

and dv(h) = d.


(iii) Let d = (d1, . . . , dn) ∈ Zn>1. If Fi is weighted homogeneous of degree di, for all i =1, . . . , n, then F is called weighted-homogeneous with respect to v with vector of degreesd.

(iv) We say that F is pre-weighted homogeneous with respect to v when qv(F )−1(0) = 0.

We refer to the works of Cima-Gasull-Mañosas [18] and Saia [71] for interesting propertiesof pre-weighted homogeneous maps.

Let v = (v1, . . . , vn) ∈ Zn>1 be a primitive vector. Let us denote by Γv+ the global Newtonpolyhedron Γ+(xv1···vn/v1

1 , . . . , xv1···vn/vnn ). We remark that Γv+ has a unique face of dimension

n − 1 not passing through the origin and that this is equal to the convex hull of the pointsbelonging to the intersection of the hyperplane of equation v1k1 + · · ·+ vnkn = v1 · · · vn withthe coordinate axis.

Lemma 2.3.6. Let F : Kn → Kn be a polynomial map. Let v ∈ Zn>1 be a primitive vectorand let d = (d1, . . . , dn) ∈ Zn>1. Then the following conditions are equivalent:

(i) F is pre-weighted homogeneous with respect to v and d = dv(F ).

(ii) F is non-degenerate with respect to Γv+ and νΓv+(Fi) = di, for all i = 1, . . . , n.

Proof. It follows easily by observing that F0(Γ+) = −v and hence the filtrating map φ :Rn

>0 → R associated to Γv+ is given by φ(k) = 〈v, k〉, for all k ∈ Rn>0.

Remark 2.3.7. Let F : Cn → Cn be a polynomial map. It is immediate to deduce from theabove definitions that, if F is Newton non-degenerate at infinity, then F is non-degeneratewith respect to Γ+(F ). An easy example showing that the converse is not true is given bythe map F : C2 → C2 defined by F (x, y) = (x+ 2y, x2 − y2).

Lemma 2.3.8. Let F : Kn → Kp be a polynomial map such that F (0) = 0 and F−1(0) iscompact. Then Γ+(F ) is convenient.

Proof. Let F1, . . . , Fp denote the component functions of F . If F is not convenient, then thereexists some i ∈ 1, . . . , n such that Γ+(Fj) does not intersect the xi-axis, for all j = 1, . . . , p.In particular we have that F vanishes on the xi-axis, since F (0) = 0, and hence F−1(0) isnot compact.

Proposition 2.3.9. Let F : Cn → Cn be a polynomial map such that F−1(0) is finite. Thenthe following conditions are equivalent:

(i) F is Newton non-degenerate at infinity,


(ii) F is non-degenerate with respect to Γ+(F ) and νΓ+(F1) = · · · = νΓ+

(Fn).

Proof. Since F−1(0) is finite, then Γ+(F ) is convenient, by Lemma 2.3.8. The implication(ii)⇒ (i) is immediate since the condition νΓ+

(F1) = · · · = νΓ+(Fn) implies that in∆(Fi) =

(Fi)∆, for all face ∆ of Γ+ such that 0 /∈ ∆ and all i = 1, . . . , n.Let us see (i)⇒ (ii). Let ∆ be a face of Γ+ of dimension n − 1 such that 0 /∈ ∆. Let

R∆ be the ring generated by the polynomials h ∈ C[x1, . . . , xn] such that supp(h) ⊆ C(∆).Since F is Newton non-degenerate at infinity, the solutions of the system (F1)∆′(x) = · · · =(Fn)∆′(x) = 0 are contained in x ∈ Cn : x1 · · ·xn = 0, for any face ∆′ of Γ+ such that∆′ ⊆ ∆. In particular, the ideal I generated by (F1)∆, . . . , (Fn)∆ in R∆ has finite colengthin R∆ by Proposition 2.2.3, which implies that I is generated by least n non-zero elementsof R∆. Then (Fi)∆ 6= 0, for all i = 1, . . . , n. In particular we have νΓ+

(F1) = · · · = νΓ+(Fn)

and thus the result follows.

It is easy to find examples showing that the above result does not hold in the case K = R.

Lemma 2.3.10. Let F = (F1, . . . , Fn) : Cn → Cn be a finite polynomial map. Let us fix aconvenient global Newton polyhedron Γ+ ⊆ Rn

+. Then the following conditions are equivalent:

(i) F is non-degenerate with respect to Γ+.

(ii) There exists a vector a = (a1, . . . , an) ∈ Zn>1 such that the map F a : Cn → Cn givenby (F a1

1 , . . . , F ann ) verifies that Γ+(F a) is homothetic to Γ+ and F a is Newton non-

degenerate at infinity.

Proof. Let us see (i)⇒ (ii). Let di = νΓ+(Fi), for all i = 1, . . . , n, and let d = d1 · · · dn. We

define ai = d/di, for all i = 1, . . . , n, and a = (a1, . . . , an). Let k ∈ Zn>0 such that k is avertex of Γ+. By condition (2.11), there exists some i ∈ 1, . . . , n such that ink(Fi) 6= 0.This means that there exists some k′ ∈ supp(Fi) such that φ(k′) = di and there exists someλ > 0 such that k′ = λk. Since di = φ(k′) = φ(λk) = λφ(k) = λM , we obtain λ = di/M . Inparticular aik′ = ai

diMk = d

Mk. Then, for any vertex k of Γ+, we have that d

Mk has integer

coordinates. This fact shows that

Γ+(F a) = d

MΓ+ = Γ+(Ad) (2.12)

Let ∆ be a face of Γ+(F a). By (2.12) there exists a face ∆′ of Γ+ such that ∆ = dM

∆′. UsingDefinition 2.3.3 it is immediate to see that (in∆′(Fi))ai = (F ai)∆ for all i = 1, . . . , n, and thuscondition (ii) follows.


Let us see the implication (ii)⇒ (i). Let a = (a1, . . . , an) ∈ Zn>1 such that Γ+(F a) and Γ+

are homothetic. Let u > 0 such that Γ+(F a) = uΓ+. Then u∆ is a face of Γ+(F a), for any face∆ of Γ+. In particular from the relation (in∆(Fi))ai = (F ai

i )u∆ for all i = 1, . . . , n, we apply (ii)and then we have that the set of solutions of the system in∆(F1)(x) = · · · = in∆(Fn)(x) = 0is contained in x ∈ Cn : x1 · · ·xn = 0.

2.4 Counting affine roots of polynomial systems

This section is devoted to show a result of Rojas and Wang [67] about the determination ofthe number of isolated affine roots, counting multiplicities, of polynomial systems. This resulttakes part of a far-reaching generalization of the result of Bernstein-Khovanskii-Kouchnirenkothat bounds the number of isolated affine roots in (C \ 0)n, counting multiplicities, ofpolynomial systems (see [2], [20], [47], [67]).

In order to state the result of Rojas and Wang we introduce some preliminary definitions.

Definition 2.4.1. Let h ∈ C[x1, . . . , xn] and let A be a compact subset of Rn>0 such that

supp(h) ⊆ A. If w ∈ Rn and h = ∑k akx

k, then we define the principal part of h withrespect to the pair (w,A), denoted by pw,A(h), as the sum of all terms akxk such that k ∈supp(h) ∩ ∆(w,A). If supp(h) ∩ ∆(w,A) = ∅, then we set pw,A(h) = 0. The polynomialpw,supp(h)(h) will be denoted simply by pw(h). Let us remark that, if h 6= 0, then pw(h) 6= 0.Moreover pw,A(h) = pw(h) if and only if ∆(w, supp(h)) ⊆ ∆(w,A). We shall refer to pw(h)as the principal part of h with respect to w.

Let A be a polytope in Zn>0, then we denote by C[A] the family of polynomials h ∈C[x1, . . . , xn] such that supp(h) ⊆ A. Let A = (A1, . . . , An) be a n-tuple of finite subsets ofZn>0. We denote by Cn[A] the set of the polynomial maps F = (F1, . . . , Fn) : Cn → Cn suchthat supp(Fi) ⊆ Ai for all i = 1, . . . , n. We can identify Cn[A] with a finite-dimensional vectorspace CN , for a sufficiently large positive integer N , by associating to each map F ∈ Cn[A]the vector formed by the coefficients of F . Under this identification, we say that a propertyholds for a generic F ∈ Cn[A] when the property holds in a dense Zariski-open subset of CN .

Definition 2.4.2. Let A = (A1, . . . , An) be an n-tuple of finite subsets of Zn>0. Let F =(F1, . . . , Fn) ∈ Cn[A]. Then we say that F is A-non-degenerate if and only if, for all w ∈ Rn

such that w0 < 0, the set of solutions of the system

pw,A1(F1)(x) = · · · = pw,An(Fn)(x) = 0


is contained in x ∈ Cn : x1 · · ·xn = 0.

Let A be a lattice polytope in Rn, then we say that A is cornered when, for all j = 1, . . . , nthere exists some k ∈ A such that kj = 0 (see [67, p. 119]). If A ⊆ Rn

>0 and fA denotes thepolynomial obtained as the sum of all terms xk such that k ∈ A ∩ Zn>0, then we observe thatA is cornered if and only if fA is not divisible by xj for all j = 1, . . . , n.

Let A = (A1, . . . , An) be an n-tuple of lattice polytopes in Rn; then A is said to becornered when Ai is cornered for all i = 1, . . . , n. We say A is nice when F−1(0) is finite fora generic map F ∈ Cn[A].

We recall that if A is a polytope in Rn, then Conv(A ∪ 0) is denoted by A0. If A =(A1, . . . , An) is an n-tuple of polytopes of Rn, then A0 = (A0

1, . . . , A0n). Let us remark that

A0 is always cornered.

Theorem 2.4.3 ([67, p. 119]). Let P = (P1, . . . , Pn) be an n-tuple of lattice polytopes of Rn>0.

Let us suppose that P is nice and cornered. Then a generic polynomial map F ∈ Cn[P] hasexactly MV(P0) roots in Cn, counting multiplicities.

The following result follows as an application of [67, Theorem 6] and is contained explicitlyin the proof of [67, Theorem 7, p. 129]. We remark that [67, Theorem 6] is actually a resultof Rojas and its proof appears in [66], moreover this is a much more general result stated forpolynomial maps over an integrally closed field of arbitrary characteristic. The proof of thisresult applies known facts about intersection theory on toric varieties.

Let A = (A1, . . . , An) be an n-tuple of finite subsets of Zn>0. Let us remark that, by[66, Proposition 8], the set of polynomial maps F ∈ Cn[A] such that F is A-non-degenerateconstitutes a dense Zariski-open subset of Cn[A].

Theorem 2.4.4. Let A = (A1, . . . , An) be an n-tuple of finite subsets of Zn>0 such thatMV(A0) > 0 and A is nice and cornered. Let F ∈ Cn[A] such that F−1(0) is finite. Then

dimCC[x1, . . . , xn]〈F1, . . . , Fn〉

6 MV(A0) (2.13)

and equality holds in (2.13) if and only if F is A0-non-degenerate.

We recall that MV(A) = 0 if and only if dim (∑i∈I Ai) < |I|, for some non-empty subsetI ⊆ 1, . . . , n, where |I| denotes the cardinal of I. In particular, this happens if |Ai| = 1,for some i ∈ 1, . . . , n.

As a corollary of Theorem 2.4.4 we obtain the following fundamental result.


Corollary 2.4.5. Let Γ+ ⊆ Rn>0 be a convenient Newton polyhedron. Let F = (F1, . . . , Fn) :

Cn → Cn be a polynomial map such that supp(Fi) ⊆ Γ+, for all i = 1, . . . , n and F−1(0) isfinite. Then µ(F ) 6 n!Vn(Γ+) and equality holds if and only if F is Newton non-degenerateat infinity.

Proof. From Definition 2.2.1 we have that F is Newton non-degenerate at infinity if and onlyif F is A-non-degenerate, with A equal to the n-tuple (Γ+ ∩ Zn>0, . . . , Γ+ ∩ Zn>0). Hence theresult follows as an immediate application of Theorem 2.4.4.

CHAPTER 3

Newton non-degeneracy at infinity

Contents3.1 Special monomials and Newton non-degeneracy at infinity . . . 44

3.2 Multiplicity and special polynomials . . . . . . . . . . . . . . . . . 53

3.3 Non-degeneracy with respect to a Newton filtration . . . . . . . 57

3.4 Newton non-degeneracy of gradient maps . . . . . . . . . . . . . . 62

3.5 Homogeneous Newton polyhedra . . . . . . . . . . . . . . . . . . . 67

In this chapter, the problem of characterizing the Newton non-degeneracy at infinitycondition for polynomial maps F : Kn → Kp using special monomials is studied.

Motivated by the characterization of the integral closure of ideals in On by analytic in-equalities proven by Lejeune-Teissier [46], we introduce the concept of special polynomialswith respect to a given polynomial map F .

The main result of this chapter (Theorem 3.1.14) characterizes the Newton non-degeneracyat infinity condition of a given polynomial map F : Kn → Kp in terms of the set of specialmonomials of the map (F, 1) : Kn → Kp+1. Moreover, when F is a convenient polynomial mapthe Newton non-degeneracy at infinity condition is characterized in terms of the set S0(F )(see Corollary 3.1.16). We remark that the results obtained in this chapter are polynomialversions of results obtained in [11, 7, 10, 33].

41

42 Chapter 3. Newton non-degeneracy at infinity

3.1 Special monomials and Newton non-degeneracy atinfinity

We begin by relating the concept of Newton non-degeneracy at infinity for polynomialmaps (see Definition 2.2.1) and the concept of Newton non-degeneracy for germs of analyticmaps (see Definition 1.2.10).

Let us fix coordinates x1, . . . , xn in Kn. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomialmap and let d = maxdeg(F1), . . . , deg(Fp). Let us introduce a new variable xn+1, so thatx1, . . . , xn, xn+1 is a coordinate system in Kn+1.

Definition 3.1.1. Let h be a polynomial in K[x1, . . . , xn], such that deg(h) 6 d. Then, wedenote by h∗ the polynomial in K[x1, . . . , xn, xn+1], defined as

h∗(x1, . . . , xn, xn+1) = xdn+1 · h(x1/xn+1, . . . , xn/xn+1). (3.1)

That is h∗ is the homogenization of degree d of h by means of the extra variable xn+1.

Naturally, the polymial map F = (F1, . . . , Fp) : Kn → Kp, induces the analytic map germF ∗ : (Kn+1, 0) → (Kp, 0), defined as F ∗ = (F ∗1 , . . . , F ∗p , xdn+1) : (Kn+1, 0) → (Kp+1, 0), whereF ∗i is the homogenization of degree d of Fi by means of the extra variable xn+1.

The following figure shows the construction in R3 of the Newton polyhedron of F ∗.

0

Г~

+

Г+( )F

Г~+F

0

( ) F

(d,0)

(0,d)(0,d,0)

(d,0,0)x1

x 2

x1

x2

*

(0,0,d)

( )

x3

Chapter 3. Newton non-degeneracy at infinity 43

Let e1, . . . , en, en+1 be the canonical basis of Rn+1, let c′ = e1 + · · ·+en+en+1 and let cdenote the projection of c′ onto the first n coordinates. Let us recall that if w = (w1, . . . , wn) ∈Rn then we denote miniwi by w0. Thus we defined Rn

0 = w ∈ Rn : w0 < 0.We now define

Rn+1∗ =

(v1, . . . , vn, vn+1) ∈ Rn+1

>0 : vn+1 = 2v0.

Next, we show that exists a bijection between the faces ∆ of Γ+(F ) not containing theorigin of Rn and the compact faces ∆∗ of Γ+(F ∗) not containing the point den+1 ∈ Rn+1.

Remark 3.1.2. We observe easily from the construction of F ∗, that any compact face ofΓ+(F ∗) is contained in the n-dimensional compact face ∆(c′,Γ+(F ∗)).

Let v = (v1, . . . , vn, vn+1) be a vector in Rn+1. Then we define the vector

w(v) = (v1 − vn+1, . . . , vn − vn+1) = (v1, . . . , vn)− vn+1c. (3.2)

We observe that if v ∈ Rn+1∗ , then the minimum of the coordinates of w(v) is equal to

−min16i6n vi, since vn+1 = 2 min16i6n vi. Therefore, we have a map w : Rn+1∗ → Rn

0 .Given a vector w = (w1, . . . , wn) ∈ Rn

0 , we define

v(w) = (w1 − 2w0, . . . , wn − 2w0,−2w0) = (w, 0)− 2w0c′. (3.3)

We observe that v(w) ∈ Rn+1∗ , for all w ∈ Rn

0 . Then we have constructed a map v : Rn0 →

Rn+1∗ . It is an easy exercise to check that v : Rn

0 → Rn+1∗ and w : Rn+1

∗ → Rn0 are bijections

and v−1 = w.Next, let us consider the map π : Rn+1 → Rn given by the projection onto the first n

coordinates i.e. π(x1, . . . , xn, xn+1) = (x1, . . . , xn), for all (x1, . . . , xn, xn+1) ∈ Rn+1. Then,from (3.2), we have that π(v) = w(v) + vn+1c, for all v ∈ Rn+1

>0 .

Lemma 3.1.3. Let us suppose that v = (v1, . . . , vn, vn+1) ∈ Rn+1>0 . Then

`(v,Γ+(F ∗)) = dvn+1 + `(w(v), Γ+(F )). (3.4)

Proof. By construction of F ∗, it is immediate to check that any element of supp(F ∗)\den+1can be written as (k, d− |k|), where k belongs to supp(F ). Then by Definition 1.1 we have

`(v,Γ+(F ∗)) = min 〈(k, kn+1), (π(v), vn+1)〉 : (k, kn+1) ∈ supp(F ∗) ∪ den+1

= min 〈(k, d− |k|), (w(v) + vn+1c, vn+1)〉 : k ∈ supp(F ) ∪ 0

= min 〈k,w(v) + vn+1c〉+ (d− |k|)vn+1 : k ∈ supp(F ) ∪ 0

= min 〈k,w(v)〉+ vn+1〈k, c〉+ (d− |k|)vn+1 : k ∈ supp(F ) ∪ 0

= dvn+1 + min 〈k,w(v)〉 : k ∈ supp(F ) ∪ 0 .


As a consequence the above result we obtain a relation among the faces of Γ+(F ) andΓ+(F ∗), as we show in the following result.

Corollary 3.1.4. Let v ∈ Rn>0. Then

∆(v,Γ+(F ∗)) =

(k, d− |k|) : k ∈ ∆(w(v), Γ+(F )).

In particular, we have that den+1 /∈ ∆(v,Γ+(F ∗)) if and only if 0 /∈ ∆(w(v), Γ+(F )).

Proof. From the definition of face, we have that (k, kn+1) ∈ ∆(v,Γ+(F ∗)) if and only if,〈(k, kn+1), (π(v), vn+1)〉 − `(v,Γ+(F ∗)) = 0. By Lemma 3.1.3, it is equivalent to

0 = 〈k, π(v)〉+ (kn+1 − d)vn+1 − `(w(v), Γ+(F )) (3.5)

= 〈k,w(v) + vn+1c〉+ (kn+1 − d)vn+1 − `(w(v), Γ+(F ))

= 〈k,w(v)〉 − `(w(v), Γ+(F )) + (k1 + · · ·+ kn+1 − d)vn+1. (3.6)

Observe that the equality (3.6) holds if and only if k1 + · · · + kn + kn+1 = d and k ∈∆(w(v), Γ+(F )), since k1 + · · · + kn + kn+1 > d and 〈k,w(v)〉 − `(w(v), Γ+(F )) > 0, for all(k, kn+1) ∈ supp(F ∗).

Let ∆ be compact face of Γ+(F )∗ not containing the point (0, . . . , 0, d) ∈ Rn+1. As aninmediate consequence of Corollary 3.1.4, we obtain that ∆ can be supported by a vector ofRn+1∗ . However ∆ could be also supported by a vector v not belonging to Rn+1

∗ .

Remark 3.1.5. Corollary 3.1.4 also shows that the projection π : Rn+1 → Rn induces abijection between the set of compact faces of Γ+(F ∗) not passing through the point den+1 ∈Rn+1 and the faces of Γ+(F ) not passing through the origin.

We also observe that, if ∆ is a compact face of Γ+(F ∗) not passing through the pointden+1 ∈ Rn+1, then

(Fi∗)∆ =((Fi)π(∆)

)∗(3.7)

for all i = 1, 2, . . . , p.It is natural to think the Newton non-degeneracy at infinity of F is related with the

Newton non-degeneracy of F ∗. The next proposition shows this fact.


Proposition 3.1.6. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. Then thefollowing conditions are equivalent:

(i) F is Newton non-degenerate at infinity.

(ii) F ∗ : (Kn+1, 0)→ (Kp+1, 0) is Newton non-degenerate.

Proof. Suppose that F ∗ is Newton degenerate, then there exists a compact face ∆ of Γ+(F ∗)not containing the point den+1 of Rn+1, such that the system

x = (x1, . . . , xn, xn+1) ∈ Kn+1 : (F ∗1 )∆(x) = · · · = (F ∗p )∆(x) = (xdn+1)∆(x) = 0, (3.8)

has a solution (q1, . . . , qn, qn+1) ∈ (K∗)n+1. Since (F ∗i )∆ is a homogeneous polynomial andqn+1 6= 0, then the system (3.8) has to the point (q1/qn+1, . . . , qn/qn+1, 1) as a solution. ByRemark 3.1.5 π(∆) is a face of Γ+(F ) not containing the origin of Rn such that (F ∗i )∆ =((Fi)π(∆)

)∗, for all i = 1, 2, . . . , p. From these equalities, it is immediate to check that

q = (q1/qn+1, . . . , qn/qn+1) is a solution of the systemx = (x1, . . . , xn) ∈ Kn : (F1)π(∆)(x) = · · · = (Fp)π(∆)(x) = 0

,

since (Fi)∗∆(q1/qn+1, . . . , qn/qn+1, 1) = (Fi)π(∆)(q1/qn+1, . . . , qn/qn+1), for all i = 1, . . . , p.Hence F is Newton degenerate at infinity.

Conversely, suppose that F is Newton degenerate at infinity, then there exists a face ∆ ofΓ+(F ) not containing the origin of Rn such that the system

x = (x1, . . . , xn) ∈ Kn : (F1)∆(x) = · · · = (Fp)∆(x) = 0, (3.9)

has a solution r = (r1, . . . , rn) ∈ (K∗)n. By Remark 3.1.5 ∆1 = π−1(∆) is a compact faceof Γ+(F ∗) not containing the point den+1 of Rn+1 and satisfies (F ∗i )∆1 = ((Fi)∆)∗, for alli = 1, . . . , p. It is immediate to check from these equalities that the point s = (r1, . . . , rn, 1) ∈(K∗)n+1 is a solution of the system

x = (x1, . . . , xn, xn+1) ∈ Kn+1 : (F ∗1 )∆1(x) = · · · = (F ∗p )∆1(x) = (xdn+1)∆1(x) = 0.

Therefore F ∗ is Newton degenerate.

Remark 3.1.7. Let F : Cn → Cn be a polynomial map. Let us suppose that F is Newtonnon-degenerate at infinity. Then we observe that, by Theorem 3.1.6 and [43, Théorème 6.2],the ideal I(F ∗) generated by the component functions of F ∗ has finite colength in the conicring A∆ = h ∈ On : supp(h) ⊆ C(∆), where ∆ is the unique compact face of Γ+(F ∗) ofdimension n.


We say that a given condition holds for all ‖x‖ 1 when there exists a constant M > 0such that the said condition holds for all x ∈ Kn for which ‖x‖ >M .

Definition 3.1.8. Let F : Kn → Kp be a polynomial map. We say that an element h ∈K[x1, . . . , xn] is special with respect to F when there exists some constant C > 0 such that

|h(x)| 6 C‖F (x)‖,

for all ‖x‖ 1.

We can consider the previous definition as a kind of global or polynomial version of thenotion of integral element over an ideal in a local ring. Let us fix coordinates x1, . . . , xn inKn. Then we define the sets:

Sp(F ) = h ∈ K[x1, . . . , xn] : h is special with respect to F ,

S0(F ) =k ∈ Zn>0 : xk is special with respect to F

.

If S0(F ) \ 0 6= ∅, then it is obvious that there exists some M > 0 such that

F−1(0) ∩ x ∈ Kn : ‖x‖ >M ⊆ x ∈ Kn : x1 · · ·xn = 0.

Proposition 3.1.9. Let F : Kn → Kp be a polynomial map and let h ∈ K[x1, . . . , xn] specialwith respect to F . Then supp(h) ⊆ Γ+(F ). In particular S0(F ) ⊆ Γ+(F ).

Proof. Suppose that h is written as h = ∑akx

k and let w = (w1, . . . , wn) ∈ Rn0 . Then we can

rewritten h ash =

∑k/∈∆

akxk +

∑k∈∆

akxk, (3.10)

where ∆ = ∆(w, h) (see Definition 2.1.7). Let us consider the meromorphic curve ϕw :K \ 0 → Kn given by ϕw(t) = (β1t

w1 , . . . , βntwn), where β = (β1, . . . , βn) ∈ (K∗)n satisfying∑

k∈∆ akβk 6= 0. Since w0 < 0, we have that limt→0 ‖ϕw(t)‖ = +∞.

By (3.10), it is easy to observe that the order of h(ϕw(t)) is given by `(w, h). If h ∈ Sp(F ),then there exists a constant C > 0 such that

|h(x)| 6 C‖F (x)‖, (3.11)

for all ‖x‖ 1. In particular, if we compose with ϕw(t) both sides of inequality (3.11) then weobtain that the limit limt→0 |h(ϕw(t))|/‖F (ϕw(t))‖ exists, which is equivalent to saying thatthe order of h(ϕw(t)) is bigger than or equal to the order of ‖F (ϕw(t))‖. That is, `(w, h) >min`(w,F1), . . . , `(w,Fp) > `(w, Γ+(F )). Therefore 〈k, w〉 > `(w, h) > `(w, Γ+(F )), for allk ∈ supp(h) and for all w ∈ Rn

0 , which means that supp(h) ⊆ Γ+(F ), by (2.3).


The next example shows that S0(F ) could be empty.

Example 3.1.10. Let F : C2 → C be the polynomial given by F (x1, x2) = x21x

22 − x1x2.

The Newton polyhedron of F at infinity is given by the segment joining the origin and thepoint (2, 2). Let γ : C∗ → C2 be the curve defined by γ(t) = (t, t−1), which is not zero andlimt→0 ‖γ(t)‖ = +∞. It is immediate to check that

1 = |xk γ(t)| > C‖F (γ(t))‖ = 0. (3.12)

for all t ∈ C∗ and for all k ∈ Γ+(F ) ∩ Z2>0. Then S0(F ) = ∅, since limt→0 ‖γ(t)‖ = +∞.

We remark that when S0(F ) 6= ∅, then it is easy to check that S0(F ) is convex. That is,if k, k′ ∈ S0(F ) then λk + (1− λ)k′ ∈ S0(F ), for all λ ∈ [0, 1] such that λk + (1− λ)k′ ∈ Zn>0.

Definition 3.1.11. Let F : Kn → Kp be a polynomial map and let h ∈ K[x1, . . . , xn]. Then,we say that h is quasi-special with respect to F when there exists some constant C > 0 suchthat

|h(x)| 6 C max‖F (x)‖, 1,

for all ‖x‖ 1. We denote by S′0(F ) the set of those k ∈ Zn>0 such that xk is quasi-specialwith respect to F .

Obviously xk is quasi-special with respect to F if and only of it is special with respect tothe map (F, 1) : Kn → Kp+1. From Proposition 3.1.9, it is immediate to check that

S0(F ) ⊆ S′0(F ) ⊆ Γ+(F ). (3.13)

The following examples show that both inclusions of (3.13) might be strict.

Example 3.1.12. Let F : C2 → C be the polynomial map given by F (x1, x2) = x21x

22. Then,

F is Newton non-degenerate at infinity, since it is monomial. The Newton polyhedron of Fat infinity is given by the segment joining the origin and the point (2, 2). Then k = (1, 1) ∈Γ+(F ). We claim that k /∈ S0(F ). Indeed, let us consider the curve γ : C∗ → C2 defined asγ(t) = (t−1, t2). It immediate to check that limt→0 ‖γ(t)‖ = +∞ and

limt→0|xk(γ(t))|/‖F (γ(t))‖ = lim

t→0|t|/|t2| = +∞.

From this, k /∈ S0(F ). However, since |x1x2| > 1 or |x1x2| 6 1, for all ‖(x1, x2)‖ > 1, weobtain that k ∈ S′0(F ). Therefore we have that S0(F ) ( S′0(F ).


Example 3.1.13. Let F = (F1, F2) : C2 → C2 be the polynomial map, whose componentfunctions are given by

F1(x1, x2) = x1 − x21x

22,

F1(x1, x2) = x2.

Observe that k = (2, 2) ∈ Γ+(F ). We claim that k /∈ S′0(F ). Indeed, let us consider themeromorphic curve γ : C∗ → C2 defined by γ(t) = (t−2, t). It is immediate to check thatlimt→0 ‖γ(t)‖ = +∞ and

limt→0|xk(γ(t))|/max‖F (γ(t))‖, 1 = lim

t→0|t−2|/max‖(0, t)‖, 1 = lim

t→0|t−2| = +∞.

From this, we have (2, 2) /∈ S′0(F ).

The next result characterizes the Newton non-degeneracy at infinity for polynomials mapsthrough of the equality S′0(F ) = k ∈ Zn>0 : k ∈ Γ+(F ).

Theorem 3.1.14. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. The followingcondition are equivalent:(i) F is a Newton non-degenerate at infinity.

(ii) S′0(F ) = Γ+(F ) ∩ Zn>0.

Proof. (i)⇒ (ii) Let us observe that F and (F, 1) have the same Newton polyhedron atinfinity. Then, by Proposition 3.1.9, it is sufficient prove that Γ+(F ) ∩ Zn>0 ⊆ S′0(F ). Letk = (k1, . . . , kn) ∈ Γ+ ∩ Zn>0, then we observe that (k1, . . . , kn, kn+1) ∈ Γ+(F ∗) ∩ Zn+1

>0 , wherekn+1 = d − |k|. By Proposition 3.1.6 we have that F ∗ : (Kn+1, 0) → (Kp+1, 0) is Newtonnon-degenerate. By Theorem 1.3.4, the monomial xk1

1 . . . xknn xkn+1n+1 is integral over the ideal of

A(Kn+1) generated by the components functions of F ∗. Therefore there exist a neighborhoodU ⊂ Kn+1 of 0 and a constant C > 0 such that

|xk11 . . . xknn x

kn+1n+1 | 6 C‖F ∗(x1, . . . , xn, xn+1)‖, (3.14)

for all (x1, . . . , xn, xn+1) ∈ U . Let M > 0 such that (e−‖x‖x1, . . . , e−‖x‖xn, e

−‖x‖) ∈ U for allx ∈ Kn such that ‖x‖ >M . Then (3.14) shows that

‖((e−‖x‖x1)k1 · · · (e−‖x‖xn)kn · (e−‖x‖)kn+1‖ 6 C‖F ∗(e−‖x‖x1, . . . , e−‖x‖xn, e

−‖x‖)‖,

for all x ∈ Cn such that ‖x‖ > M . Thus, if we multiply both sides of the last inequality by|e‖x‖.d|, we have that

‖xk‖ 6 C max‖F (x)‖, 1, (3.15)


for all ‖x‖ 1. Therefore S′0(F ) = Γ+(F ) ∩ Zn>0.(ii) ⇒ (i) By contradiction, if we suppose that F is Newton degenerate at infinity, then

by Lemma 2.1.5, there exists a primitive vector w = (w1, . . . , wn) ∈ Rn0 such that if ∆ =

∆(w, Γ+), then the system

(F1)∆(x) = (F2)∆(x) = · · · = (Fp)∆(x) = 0,

has a solution q = (q1, . . . , qn) ∈ (K∗)n. In particular the curve γ : K \ 0 → Kn defined byγ(t) = (q1t

w1 , . . . , qntwn) is not zero and limt→0 ‖γ(t)‖ =∞, since w0 < 0.

Let r = `(w, Γ+), which is a negative number, by Lemma 2.1.5. Since (Fi)∆ is a weightedhomogeneous polynomial with respect to w, we obtain

(Fi)∆(λw1x1, . . . , λwnxn) = λr(Fi)∆(x1, . . . , xn), (3.16)

for all i = 1, 2, . . . , p and for all λ ∈ K∗. By the equality (3.16) we have

F (λw1x1, . . . , λwnxn) = λrF∆(x1, . . . , xn) + FR(λw1x1, . . . , λ

wnxn), (3.17)

for all λ ∈ K∗, where F∆ and FR are the polynomial maps given by:

F∆ = ((F1)∆, . . . , (Fp)∆) .

FR = (F1 − (F1)∆, . . . , Fp − (Fp)∆) .

Now, let us consider a vertex s = (s1, . . . , sn) ∈ Zn>1 of ∆. We claim that s /∈ S′0(F ).Indeed, if we use (3.16), (3.17) and F∆(q) = 0, we obtain that

limt→0

|xs(γ(t)|‖(F (γ(t)), 1)‖ = lim

t→0

|qs11 · · · qsnn t〈s,w〉|

‖(trF∆(q) + FR(γ(t)), 1)‖ = limt→0

|qstr|‖(FR(γ(t)), 1)‖ = +∞, (3.18)

since the order of ‖(FR(γ(t)), 1)‖ is bigger than r. Therefore (3.18) shows that s /∈ S′0(F ),which shows that S′0(F ) 6= Γ+(F ) ∩ Zn>0.

The following example shows that the Newton non-degeneracy at infinity condition is notsufficient to obtain the equality S0(F ) = S′0(F ).

Example 3.1.15. Let us consider the polynomial map F = (F1, F2) : C2 → C2, whose thecomponent functions are given by

F1(x1, x2) = x1x32,

F2(x1, x2) = x31x

22.


Since F has monomial components, F is Newton non-degenerate at infinity. Moreover, since(1, 1) ∈ Γ+(F ), the monomial h = x1x2 is quasi-special, by Theorem 3.1.14.

However, we claim that h = x1x2 /∈ Sp(F ). Indeed, let us consider the meromorphic curveγ : C∗ → C2 defined by γ(t) = (t−1, t3). It is immediate to check that limt→0 ‖γ(t)‖ = +∞,limt→0 ‖F (γ(t))‖ = limt→0 ‖(t8, t3)‖ = 0 and

limt→0

‖F (γ(t))‖|h(γ(t))| = lim

t→0‖(t6, t)‖ = 0.

The claim is an immediate consequence of this fact.

In the next result, we characterize the Newton non-degeneracy condition for convenientpolynomial maps through of the equality S0(F ) = S′0(F ) = xk : k ∈ Γ+(F ). We remarkthat this result is the polynomial version of a result proved by Saia (see Theorem 1.2.5).

Corollary 3.1.16. Let F = (F1, . . . , Fp) : Kn → Kp be a convenient polynomial map. Thenthe following conditions are equivalent:

(i) F is Newton non-degenerate at infinity.(ii) S0(F ) = S′0(F ) = Γ+(F ) ∩ Zn>0.

Proof. (i)⇒ (ii) By Proposition 3.1.9 it is sufficient prove that Γ+(F ) ∩ Zn>0 ⊆ S0(F ). ByTheorem 3.1.14, there exists a constant C > 0 such that

‖xk‖ 6 C‖(F (x), 1)‖, (3.19)

for all ‖x‖ 1 and for all k ∈ Γ+(F ) ∩ Zn.Let ri = maxr > 0 : rei ∈ Γ+(F ), for all i = 1, . . . , n and r0 = minr1, . . . , rn. Since

F is convenient, we have that‖xrii ‖ 6 C ′‖(F (x), 1)‖, (3.20)

for some constant C ′ > 0, for all ‖x‖ 1 and for all i = 1, . . . , n. Then as an immediateconsequence of these inequalities we obtain

‖x‖r0(F ) 6 C ′′‖(F (x), 1)‖, (3.21)

for all ‖x‖ 1, for some constant C ′′ > 0. From the inequality (3.21), it is immediate to checkthat ‖F (x)‖ > 1, for all ‖x‖ 1 and (3.19), shows that k ∈ S0(F ), for all k ∈ Γ+(F ) ∩ Zn.Therefore S0(F ) = S′0(F ) = Γ+(F ) ∩ Zn>0.

(ii)⇒ (i) It is immediate consequence from Theorem 3.1.14, since S0(F ) = S′0(F ) =Γ+(F ) ∩ Zn>0.


Corollary 3.1.17. Let F = (F1, . . . , Fp) : Kn → Kp be a convenient polynomial map. If Fis Newton non-degenerate at infinity, then F is proper.

Proof. Since Γ+(F ) is convenient, the result is an immediate application of Corollary 3.1.16.

3.2 Multiplicity and special polynomials

Let us consider I an ideal in a Noetherian local ring (R,m) with infinite residue fieldk, dim(R) = d. Suppose that I is m-primary and denote by e(I) the multiplicity of I, inthe sense of Samuel introduced in Chapter 1. If I is generated minimally by d elementsg1, . . . , gd ∈ R and h ∈ R, then by virtue of a result of Northcott-Rees, Theorem 1.1.6, andRees’ Theorem, the multiplicity e(I + 〈h〉) can be expressed as

e(I + 〈h〉) = e(g1 + α1h, . . . , gd + αdh)

for a generic choice of (α1, . . . , αd) ∈ kd. Moreover, e(I + 〈h〉) = e(I) if and only if h isintegral over I. Since, we have a notion of element integer for polynomial maps, the aboveresult motivates the study of the polynomial version of this result.

This section is devoted to the characterization of special polynomials with respect toa given finite polynomial map F = (F1, . . . , Fn) : Cn → Cn in terms of its multiplicity.Recall from Chapter 1, that F + hα : Cn → Cn denotes the polynomial map defined asF + αh = (F1 + α1h, . . . , Fn + αnh), where α = (α1, . . . , αn) ∈ Cn and h ∈ C[x1, . . . , xn].

Lemma 3.2.1. Let F : Cn → Cn be a finite polynomial map and h ∈ C[x1, . . . , xn]. Thenthere exist positive constants M and δ, such that ‖F (x)‖ 6= 0 for all ‖x‖ >M , and

‖(F + hα)(x)‖ 6= 0, for all x ∈ Cn with ‖x‖ = M, (3.22)

for all α ∈ Cn such that ‖α‖ < δ.

Proof. Let CM = sup|h(x)| : ‖x‖ = M, since h is not the zero polynomial and F : Cn → Cn

is a finite polynomial map, there exists M > 0 such that CM > 0 and ‖F (x)‖ 6= 0 for allx ∈ Cn, such that ‖x‖ > M . Let us consider δ1 = min‖F (x)‖ : ‖x‖ = M. It is immediateto see that δ1 > 0, since F (x) 6= 0 for all ‖x‖ = M .

Let δ = δ12CM . Then, for all α ∈ Cn, such that ‖α‖ < δ, we obtain that

‖(F + hα)(x)‖ > ‖F (x)‖ − ‖α‖|h(x)| > δ1 −δ1

2 = δ1

2 > 0,

for all x ∈ Cn, with ‖x‖ = M .


Next we show a result that characterizes the special polynomials with respect to a givenfinite polynomial map F : Cn → Cn in terms of multiplicity.

Theorem 3.2.2. Let F : Cn → Cn be a finite polynomial map and let h be a polynomialfunction. Then h is special with respect to F if and only if, there exists δ > 0 such that(i) F + hα is a finite polynomial map and

(ii) µ(F ) = µ(F + hα),for all α ∈ Cn such that ‖α‖ < δ.

Proof. Let us suppose that h is special with respect to F , then since #F−1(0) is finite, thereexist positive constants C and M such that

|h(x)| 6 C‖F (x)‖ and ‖F (x)‖ 6= 0, for all ‖x‖ >M.

Let us consider δ = 1C> 0. Then, an immediate computation shows that for any α ∈ Cn

such that ‖α‖ < δ,‖(F + hα)(x)‖ > ‖F (x)‖(1− ‖α‖C) > 0, (3.23)

for all x ∈ Cn such that ‖x‖ >M . Then, we have that (3.23) implies that F +hα : Cn → Cn

is a finite polynomial map, for all ‖α‖ < δ, since every bounded algebraic set in Cn is finite.Now, let us fix α ∈ Cn, such that ‖α‖ < δ and consider the homotopyH : Cn×[0, 1]→ Cn,

given by H(x, t) = (F + htα)(x). Then, by (3.23) we obtain that H satisfies the followingconditions:(i) Ht : Cn → Cn is a finite polynomial map for all t ∈ [0, 1].

(ii) Ht(x) 6= 0, for all ‖x‖ >M and for all t ∈ [0, 1].

(iii) H(x, 0) = F (x) and H(x, 1) = (F + hα)(x), for all x ∈ Cn.The above conditions imply that µ(F ) = µ(F +hα) as a consequence of the invariance of theindex by homotopies Theorem 1.4.7 and the equality between the index and the codimensionof finite ideals in C[x1, . . . , xn] (see for instance [18] and [20]).

Conversely, let us suppose that h is not special with respect to F . Then, it follows fromdefinition there exists a sequence xmm>1 in Cn such that

h(xm) 6= 0, for all m > 0, limm→∞

‖xm‖ = +∞ and limm→∞

‖F (xm)‖|h(xm)| = 0. (3.24)

Since F is a finite polynomial map, by Lemma 3.2.1 we choose positive constants M and δ1,such that ‖F (x)‖ 6= 0 for all ‖x‖ >M and

‖(F + hα)(x)‖ 6= 0, for all x ∈ Cn with ‖x‖ = M, (3.25)


for all α ∈ Cn such that ‖α‖ < δ1. By (3.24) there exists m0 such that ‖xm0‖ > M and‖F (xm0)‖/|h(xm0)| < minδ, δ1. Thus we can consider the polynomial map F + hα0 : Cn →Cn, where α0 = −F (xm0)/h(xm0). Suppose that F + hα0 is a finite polynomial map. Thenby (3.25) the homotopy H : Cn × [0, 1] → Cn, defined by H(x, t) = (F + htα0)(x) satisfiesthe following conditions:

(i) H(x, 0) = F (x) and H(x, 1) = (F + hα0)(x), for all x ∈ Cn.

(ii) H(x, t) 6= 0, for all ‖x‖ = M and for all t ∈ [0, 1].

Let us denote by U the open set x ∈ Cn : ‖x‖ < M. Then, by Theorem 1.4.7, we obtainthat

ind(F,U) =∑

x∈F−1(0)∩Uindx(F ) =

∑y∈(F+hα0)−1(0)∩U

indy(F + hα0) = ind(F + hα0, U). (3.26)

But (3.25), shows that F−1(0) ⊂ U and a direct computation shows that (F +hα0)(xm0) = 0,with xm0 /∈ U . Therefore (3.26) shows that µ(F ) < µ(F + hα0).

Example 3.2.3. Let us consider the polynomial map F = (F1, F2) : Cn → Cn, whosecomponent functions are given by F1(x1, x2) = x1 +x1x

32 and F2(x1, x2) = x3

2. It is immediateto check that F is a finite polynomial map and µ(F ) = 3.

Let h be the polynomial given by h = x1x32. It is immediate to check that F+hα : C2 → C2

is a finite polynomial, for all α = (α1, α2) ∈ C2. By Lemma 3.2 given in [43], we obtain that

µ(F + hα) = 3 + dimCC[x1, x2]

〈1 + x32(1 + α1), 1 + α2x1〉

.

From this equality is easy to see that µ(F + hα) = 6 for all α ∈ α = (α1, α2) ∈ C2 : α1 6=−1 or α2 6= 0. Therefore, by Theorem 3.2.2 h /∈ Sp(F ).

The following result characterizes the Newton non-degeneracy at infinity condition forfinite polynomial maps Cn → Cn.

Corollary 3.2.4. Let F : Cn → Cn be a finite polynomial map. Then the following conditionsare equivalent:

(i) F is Newton non-degenerate at infinity.

(ii) µ(F ) = n!Vn(Γ+(F )).

(iii) S0(F ) = Γ+(F ) ∩ Zn>0.


Proof. It is immediate consequence of Corollaries 2.4.5 and 3.1.16.

In the following result, let us consider F : Cn → Cn a finite polynomial, from thispolynomial map, consider the corresponding analytic map germ F ∗ : (Cn+1, 0) → (Cn+1, 0),given in the above section. Then we observe that, F is Newton non-degenerate at infinity,if and only if F ∗ is Newton non-degenerate (Theorem 3.1.6) and by Remark 3.1.7 the idealI(F ∗) generated by the components functions of F ∗ has finite colength in the conic ringA∆ = h ∈ On : supp(h) ⊆ C(∆), where ∆ is the unique compact face of Γ+(F ∗) ofdimension n. Therefore, it makes sense to compute the multiplicity in the sense of Samuel ofI(F ∗) in A∆, since A∆ is a local ring. We denote this multiplicity by e(I(F ∗),A∆).

From the construction of F ∗, it is immediate to see that Γ−(F ∗) is an (n+ 1)-dimensionalpolytope with basis ∆ contained in Rn+1

>0 and height h = d/√n+ 1. Thus we can consider

the (n + 1)-dimensional volume of Γ−(I(F ∗)). To calculate the (n + 1)-dimensional volumeof Γ+(F ∗), observe that the projection onto the first n coordinates of ∆, is the Newtonpolyhedron Γ+(F ) i.e π(∆) = Γ+(F ) and this fact shows that Vn(∆) =

√n+ 1 ·Vn(Γ+(F )).

By integration, we obtain that Vn+1(Γ−(F ∗)) = Vn(∆) · h/(n + 1). Therefore the aboverelations show that

Vn+1(Γ−(F ∗)) = d

n+ 1Vn(Γ+(F )). (3.27)

In the next result, we give a relation among the multiplicity of F and the multiplicitye(I(F ∗),A∆), when n = p.

Corollary 3.2.5. Let F = (F1, . . . , Fn) : Cn → Cn be a finite polynomial map. ThenF is Newton non-degenerate at infinity if and only if e(I(F ∗),A∆) = dµ(F ), where d =maxdeg(F1), . . . , deg(Fn).

Proof. The result follows of the equality (3.27) and as an immediate consequence of Theorem1.2.9, Proposition 3.1.6 and Corollary 3.2.4.

Example 3.2.6. Let a, bi, ci,∈ Z3>1, for all i = 1, 2, 3 such that b1 < b2 < b3 and c1 < c2 < c3.

Let us consider the polynomial map F = (F1, F2, F3) : C3 → C3, whose component functionsare given by

F1(x1, x2, x3) = xa1 + xb12 x

c33 + xb3

2 xc13 ,

F2(x1, x2, x3) = xb22 + xb1

2 xc33 + xb1

2 xc23 ,

F3(x1, x2, x3) = xc23 + xb3

2 xc13 + 2xb2

2 xc13 .


The Figure 3.1 shows the picture of Γ+(F ). From Definition 2.2.1, it is immediate to seethat F is Newton non-degenerate at infinity. Therefore, by Corollary 3.2.4 and a directcomputation on the volume of Γ+(F ), we obtain that

µ(F ) = dimC C[x1, x2, x3]/I(F ) = a · (b1c2 + b2c1 + b3c3 − b1c1).

x1

x2

x3

b 2

1 3

0

Γ~+( )F

(0, , )

(0, , )3 1

(0,0, )c2

b c

b c

( ,0,0)a

(0, ,0)

Figure 3.1: Newton polyhedron of F .

3.3 Non-degeneracy with respect to a Newton filtration

In this section we compute the multiplicity of finite polynomial maps Cn → Cn thatsatisfies some non-degeneracy conditions introduced in Chapter 2.

Our first goal is to determine the multiplicity of a finite polynomial map Cn → Cn, non-degenerate with respect to a fixed global Newton polyhedron. We remark that the followingresult is the polynomial version of Theorem 1.2.12.

Let us fix a Newton polyhedron at infinity Γ+ ⊆ Rn>0 and νΓ+

: C[x1, . . . , xn]→ Z∪−∞denotes the Newton filtration associated to Γ+.

Corollary 3.3.1. Let F = (F1, . . . , Fn) : Cn → Cn be a finite polynomial map, then

µ(F ) 6νΓ+

(F1) · · · νΓ+(Fn)

Mnn!Vn(Γ+) (3.28)

and equality holds if and only if the map F is non-degenerate with respect to Γ+, where Mdenotes the value of νΓ+

along the monomials with exponents on Γ.


Proof. As in the proof of Lemma 2.3.10, let di = νΓ+(Fi), for all i = 1, . . . , n, d = d1 · · · dn and

a = (a1, . . . , an) ∈ Zn>1, where ai = d/di, for all i = 1, . . . , n. Let us consider the polynomialmap F a : Cn → Cn defined as F a = (F a1

1 , . . . , F ann ). It is immediate to see that F is finite if

only if F a is finite, since F−1(0) = (F a)−1(0). Then by [43, Lemma 3.2] and Corollary 3.2.4,we obtain that

µ(F ) = d1 · · · dndn

µ(F a) 6 d1 · · · dndn

n!Vn(Γ+(F a)). (3.29)

The inequality (3.28) follows from (3.29), since Γ+(F a) = (d/M)Γ+ (see Lemma 2.3.10).Now, suppose that F is non-degenerate at infinity with respect to Γ+, then by Lemma

2.3.10 we have that F a is Newton non-degenerate at infinity. Thus the inequality (3.29) turnsinto equality, which implies the equality in (3.28).

Conversely, suppose that µ(F ) = (d1 · · · dn/Mn)n!Vn(Γ+). From the inequality (3.29)we obtain that µ(F a) = n!Vn(Γ+(F a)), since Γ+(F a) = (d/M)Γ+. Therefore by Corollary3.2.4 and Lemma 2.3.10, the polynomial map F is non-degenerate at infinity with respect toΓ+.

Example 3.3.2. Let F = (F1, F2) : C2 → C2 be the polynomial map whose componentfunctions are given by

F1(x1, x2) = x41 + x6

1x62 + x6

2 + α,

F2(x1, x2) = x42 + x4

1x42 + β.

Where α, β ∈ C. Let g be the polynomial defined by g(x1, x2) = x41 + x6

1x62 + x6

2 an letΓ+ = Γ+(g). Then we observe that Γ+ is convenient and F0(Γ+) = −e2, (−3, 1).

A direct computation shows that the Newton filtration map φΓ+: R2

>0 → R2 is given byφΓ+

((k1, k2)) = max2k2, 3k1 − k2, from this fact we obtain that M = 12, νΓ+(F1) = 12

and νΓ+(F2) = 8. It is straightforward to check that F is Newton degenerate at infinity.

However, F is non-degenerate with respect to Γ+. Therefore, by Corollary 3.3.1, we obtainthat µ(F ) = (νΓ+

(F1) · νΓ+(F2)/M2)2!V2(Γ+) = 40.

Let v ∈ Zn>1 a primitive vector and let h ∈ C[x1, . . . , xn]. Let us suppose that h is writtenas h = ∑

akxk. Then we recall from Section 2.3 that dv(h) is defined by dv(h) = max〈v, k〉 :

k ∈ supp(h), and the principal part of h at infinity with respect to v, denoted by qv(h) isdefined as the sum of those akxk, such that 〈v, k〉 = dv(h).

Now, we show the polynomial version of the Arnold’s formula for weighted homogeneousmap germs, see [11, Corollary 3.4]. We determine the multiplicity of a finite polynomial map


Cn → Cn, non-degenerate with respect to a fixed global Newton polyhedron with only one(n− 1)-dimensional face, determined by a fixed primitive vector v = (v1, . . . , vn) ∈ Zn>1.

Corollary 3.3.3. Let F = (F1, . . . , Fn) : Cn → Cn be a finite polynomial map. Let us fix avector v = (v1, . . . , vn) ∈ Zn>1 and let di = dv(Fi), for all i = 1, . . . , n. Then

µ(F ) 6 d1 · · · dnv1 · · · vn

and equality holds if and only if the map qv(F ) is finite.

Proof. Let e1, . . . , en the canonical basis in Rn and consider the convenient global Newtonpolyhedron at infinity Γ+ determined by the points v−1

i ei : i = 1, . . . , n. The Newtonfiltration map φ associated to Γ+ is given by φ(k1, . . . , kn) = v1k1 + · · · + vnkn , for allk = (k1, . . . , kn) ∈ Rn

>0. Thus νΓ+(Fi) = di, for all i = 1, . . . , n and M = 1. Then if we use

Corollary 3.3.1, we obtain

µ(F ) 6 d1 · · · dnn!Vn(Γ+) = d1 · · · dnv1 · · · vn

,

and equality holds if and only if F is non-degenerate with respect to Γ+.From Lemma 2.3.10, for any face of ∆ of Γ+(F a) there exists a face ∆′ of Γ+ such that

∆ = d∆′, where d = d1 · · · dn and it is immediate to check that

(F aii )∆ = (in∆′(Fi))ai = (in∆′(qv(Fi))ai = ((qv(Fi))ai)∆ , (3.30)

for all i = 1, . . . , n and for all face ∆ of Γ+(F a) not containing the origin.From (3.30), it follows that F a is Newton non-degenerate at infinity if and only if the

polynomial map qv(F )a : Cn → Cn defined as qv(F )a = ((qv(F1))a1 , . . . , (qv(Fn))an) is Newtonnon-degenerate at infinity. However, by Theorem 2.2.3 we have that the map qv(F )a isNewton non-degenerate at infinity if and only if qv(F )a is a finite polynomial map. Thereforeby Lemma 2.3.10, we have that F is Newton non-degenerate with respect to Γ+ if and onlyif qv(F )a : Cn → Cn is a finite polynomial map. The result follows, since

(qv(F )a

)−1(0) =(

qv(F ))−1

(0).

Example 3.3.4. Let v = (2, 3, 2) ∈ Z3>1. Let F = (F1, F2, F3) : C3 → C3 be the polynomial

map whose component functions are given by

F1(x1, x2, x3) = x21 − x2

1x22 + x2

3 + 1,

F2(x1, x2, x3) = x32 + x2

1x22x3 − x4

3,

F3(x1, x2, x3) = x21x

22 − 1.


It is immediate to see that F is a finite polynomial map. A direct computation shows thatdv(F ) = (10, 12, 10) and the principal part of F at infinity with respect to v is given byqv(F ) = (−x2

1x22, x

21x

22x3, x

21x

22). We observe that qv(F ) is not finite, then a simple computa-

tion using Singular [27] shows that µ(F ) = 28 < 10·12·10/2·3·2 = 100. On the other hand,let us consider the polynomial map G = (G1, G2, G3) : C3 → C3 such that G1 = F1 +x5

1 +x53,

G2 = F2 +x42 +2x6

3 and G3 = F3. Then it is immediate to check that qv(G) = 0. Therefore,by Corollary 3.3.3, we have µ(G) = 100.

The next definition is a polynomial version for the class of pre-weighted homogeneousfunction germs, given in [71].

Definition 3.3.5. Let f : Cn → C be a polynomial function. Let us fix a vector v ∈ Zn>1.Then f is called pre-weighted homogeneous with respect to v if and only if the gradient map∇qv(f) : Cn → Cn is a finite polynomial map, that is, when qv(f) has at most a finite numberof singular points.

If f : Cn → C is a complex polynomial function with a finite number of singular points,and J(f) denotes the ideal of C[x1, . . . , xn] generated by ∂f

∂x1, . . . , ∂f

∂xn, then theMilnor number

at infinity of f , denoted by µ∞(f), is defined as

µ∞(f) = dimCC[x1, . . . , xn]

J(f) .

Example 3.3.6. Let v ∈ Zn>1 be a primitive vector. Then the following examples show thatthe condition dv( ∂f∂xi ) = d− vi, for all i = 1, . . . , n, or that qv(∇f) is a finite polynomial mapare not sufficient conditions for f to be pre-weighted homogeneous with respect to v.

(i) Let v = (2, 1) and let f be the polynomial function given by f(x1, x2) = x32+x1x2+x1x

22.

Then, it is straightforward to check that f has a a finite number of singular points anddv( ∂f∂xi ) = d − vi for i = 1, 2, where d = dv(h) = 4. However, f is not pre-weightedhomogeneous with respect to v, since qv(f) = x1x

22 does not have a finite number of

singular points.

(ii) We consider the polynomial f given by f = x31 +x2

2. Then a direct computations showsthat f has a a finite number of singular points and qv(∇f) = (3x2

1, 2x2) is a finitepolynomial map. However, as we can be seen f is not pre-weighted homogeneous withrespect to v, since qv(f) = x3

1 does not have a finite number of singular points.

Next, we characterize the pre-weighted homogeneous polynomials.


Lemma 3.3.7. Let f : Cn → C be a polynomial function with a finite number of singularpoints. Let us fix a vector v ∈ Zn>1 and let d = dv(f). Then f is pre-weighted homogeneouswith respect to v if and only if satisfies the following conditions

(i) dv( ∂f∂xi ) = d− vi, for all i = 1, . . . , n.

(ii) qv(∇f) : Cn → Cn is a finite polynomial map.

Proof. Let A denotes the support of f and let Ai = k ∈ A : ki > 0, for all i = 1, . . . , n.Then we have

dv( ∂f∂xi

)= max〈v, k − ei〉 : k ∈ Ai = max〈v, k〉 : k ∈ Ai − vi, (3.31)

for all i = 1, . . . , n.Suppose that f is pre-weighted homogeneous with respect to v, then we would have that

for all i ∈ 1, . . . , n, there exists k = (k1, . . . , kn) ∈ supp(qv(f)) such that ki > 0. From this,it is straightforward to check that dv( ∂f∂xi ) = d− vi, for all i = 1, . . . , n. Also, we can observethat if dv( ∂f∂xi ) = d− vi, then by the equality given in (3.31) it is immediate to check that thefollowing equality holds

∂

∂xiqv(f) = qv

(∂f

∂xi

). (3.32)

Therefore, from (3.32) it follows that qv(∇f) : Cn → Cn is a finite polynomial map.Conversely, since the conditions (i) and (ii) are satisfied, we have that the equality in (3.32)

holds. For this, it is immediate to see that f is pre-weighted homogeneous with respect tov.

Next, we give a result which is the polynomial version of the main result given by Milnor-Orlik in [55] (see also M. Furuya and M. Tomari [33]).

Corollary 3.3.8. Let f : Cn → C be a polynomial function with a finite number of singularpoints. Let us fix a vector v ∈ Zn>1 and let d = dv(f). Then

µ∞(f) 6 (d− v1) · · · (d− vn)v1 · · · vn

(3.33)

and equality holds if and only if f is pre-weighted homogeneous.

Proof. Since that f : Cn → C is a polynomial function with a finite number of singular points,then the polynomial map ∇f given by ( ∂f

∂x1, . . . , ∂f

∂xn) : Cn → Cn is finite. If we take the same

Newton polyhedron taken in the proof of Corollary 3.3.3, then we have νΓ+( ∂f∂xi

) = dv( ∂f∂xi ),


for all i = 1, . . . , n. Let di = dv( ∂f∂xi ), then a direct computation shows that di 6 d − vi, forall i = 1, . . . , n. From Corollary 3.3.3, we have that

µ∞(f) = µ(F ) 6 d1 · · · dnv1 · · · vn

6(d− v1) · · · (d− vn)

v1 · · · vn

and the equality in the first inequality holds if and only if qv(∇f) is a finite polynomial map.It is easy to observe that the equality (3.33) holds if and only if dv( ∂f∂xi ) = d − vi for alli = 1, . . . , n. and qv(∇f) is a finite polynomial map. Therefore, by Lemma 3.3.7 we havethat equality (3.33) holds if and only if f is pre-weighted homogeneous.

3.4 Newton non-degeneracy of gradient maps

Let f : Cn → C be a polynomial function. We denote by ∇f the gradient map of f .That is, ∇f is the map Cn → Cn given by ∇f =

(∂f∂x1, . . . , ∂f

∂xn

). In this section we study the

relation between polynomials f ∈ C[x1, . . . , xn] which are Newton non-degenerate at infinity(see Definition 2.2.7) and polynomials of C[x1, . . . , xn] whose gradient map is Newton non-degenerate at infinity. This study is motivated by the results given in [7] for the case offunction germs (Cn, 0)→ C.

Let I ⊆ 1, . . . , n, I 6= ∅. Then we define KnI = (x1, . . . , xn) ∈ Kn : xi = 0, for all i /∈

I. We denote the cardinal of I by |I|. If f = ∑k akx

k is a polynomial in C[x1, . . . , xn], thenfI denotes the sum of all terms akxk such that k ∈ supp(f)∩Rn

I . If no such terms exist, thenwe set fI = 0.

We recall that, if f ∈ C[x1, . . . , xn], then A(f) denotes the polynomial map Cn → Cn

given byA(f) =

(x1∂f

∂x1, . . . , xn

∂f

∂xn

).

Let Γ+ ⊆ Rn be a Newton polyhedron at infinity. Then we denote by ϑ(Γ) the set ofvertices of Γ+ and by fΓ the polynomial obtained as the sum of all monomials xk such thatk ∈ ϑ(Γ). Let ∆ be a face of Γ+, we say that ∆ is simplicial when ∆ is equal to the convexhull of dim(∆) + 1 vertices of Γ+. Then, we will say that Γ+ is simplicial when any face ofΓ+ not passing through the origin is simplicial.

Lemma 3.4.1. Let Γ+ be a convenient Newton polyhedron in Rn>0. Let us suppose that

Γ+ is simplicial. Then fΓ is Newton non-degenerate at infinity and the polynomial mapA(fΓ) : Cn → Cn is finite.


Proof. Let k1, . . . , ks be the set of vertices of Γ+. Let ∆ be a face of Γ+, suppose that ∆is equal to the convex hull of k1, . . . , kr for some r 6 s. Since ∆ is simplicial, we have thatk1, . . . , kr are linearly independent and r = dim ∆ + 1. In order to see that fΓ is Newtonnon-degenerate at infinity, by definition it is enough to check that

x ∈ Cn :(x1∂fΓ∂x1

)∆

(x) = · · · =(xn∂fΓ∂xn

)∆

(x) = 0⊆ x ∈ Cn : x1 · · ·xn = 0 .

We observe that (xi∂fΓ∂xi

)∆

= xi

(∂(fΓ)∆

∂xi

)for all i = 1, . . . , n. Therefore, if we write ki = (k1

i , . . . , kni ) for i = 1, . . . , r, the above system

of equations is translated into

k11x

k1 + · · ·+ k1rx

kr = 0,

· · ·

kn1xk1 + · · ·+ knr x

kr = 0.

Since k1, . . . , kr are linearly independent, then xki = 0 for all i = 1, . . . , n. But this meansthat some coordinate of x must be zero. Thus fΓ is Newton non-degenerate at infinity andby Corollary 3.1.17 the polynomial map A(fΓ) : Cn → Cn is finite.

Corollary 3.4.2. Let F : Cn → Cn be a finite polynomial map, such that Γ+(F ) is simplicial.Then F is Newton non-degenerate at infinity if and only if

µ(F ) = µ(A(fΓ)).

Proof. Let Γ+ = Γ+(F ). The function fΓ is Newton non-degenerate at infinity by Lemma3.4.1, since Γ+ is simplicial. Then the multiplicity of A(fΓ) is equal to n!Vn(Γ+(fΓ)). ButΓ+ = Γ+(fΓ). Hence the number n!Vn(Γ+) is equal to the multiplicity of A(fΓ) and the resultfollows from Corollary 3.2.4.

Let Γ+ ⊆ Rn>0 be a convenient and simplicial Newton polyhedron. As a consequence of

the above corollary, if f : Cn → C is a polynomial function such that A(f) is finite andΓ+(f) = Γ+, then f is Newton non-degenerate at infinity if and only if

µ(A(f)) = µ(A(fΓ)). (3.34)

We remark that the dimensions appearing on both sides of (3.34) are easy computable viaa computer algebra system such as Singular [27]. We also point out that, given a (local)


Newton polyhedron Γ+ ⊆ Rn>0, the program Gérmenes [56] developed by A. Montesinos

provides the list of compact faces of Γ+ and the vertexes of that faces, whenever n 6 8.Then, with the aid of this program and the correspondence indicated in Corollary 3.1.4, it ispossible to determine when a global Newton polyhedron Γ+ ⊆ Rn

>0 is simplicial if n 6 7.We remark that the next result is the polynomial version of Theorem 5.2 shown by Bivià-

Ausina in [7].

Theorem 3.4.3. Let f : Cn → C be a polynomial function with a finite number of singularpoints such that

∂fI

∂xi=(∂f

∂xi

)I

, (3.35)

for all I ⊆ 1, . . . , n and all i = 1, . . . , n. Suppose that the polynomial map A(f) : Cn → Cn

is finite. If ∇f is Newton non-degenerate at infinity , then f is also Newton non-degenerateat infinity.

Proof. Let Γ+ = Γ+(f). We observe that Γ+ is convenient, since A(f) is a finite polynomialmap. By [43, Lemma 3.2], we know that

µ(A(f)) =∑

I⊆1,...,nµ∞(fI) (3.36)

where we set µ∞(f∅) = 1 and µ∞(f1,...,n) = µ∞(f). Moreover, since the polynomial map∇f : Cn → Cn is Newton non-degenerate at infinity and by condition (3.35), we deducethat the polynomial map ∇fI : Cn

I → CnI is also Newton non-degenerate at infinity (as a

polynomial map), for all I ⊆ 1, . . . , n. Thus applying Theorem 3.2.4 and relation (3.36),we have

n!Vn(Γ+) > µ(A(f)) =∑

I⊆1,...,n|I|!V|I|(Γ+(∇fI)). (3.37)

Since the Newton non-degeneracy at infinity is a generic condition (see [43, p. 26]), thenit is possible to find a polynomial function g ∈ C[x1, . . . , xn] satisfying that g is Newton non-degenerate at infinity and supp(f) = supp(g). In particular, we have Γ+(∇gI) = Γ+(∇fI)and we deduce the following chain of inequalities:

n!Vn(Γ+) = µ(A(g)) =∑

I⊆1,...,nµ∞(gI) 6

∑I⊆1,...,n

|I|!V|I|(Γ+(∇(gI))

=∑

I⊆1,...,n|I|!V|I|(Γ+(∇fI)) =

∑I⊆1,...,n

µ∞(fI) = µ(A(f)) 6 n!Vn(Γ+).

Then, we deduce that n!Vn(Γ+) is equal to µ(A(f)) and hence the polynomial function f isNewton non-degenerate at infinity, by Corollary 3.2.4.


Next we show that condition (3.35) cannot be removed from Theorem 3.4.3.

Example 3.4.4. Let f : C3 → C be the polynomial function given by f(x1, x2, x3) =x2

1 + x22 + x4

3 + x1x2(x1 − x2)2 + x32x3. We observe that f is Newton degenerate at infinity,

since the function f1,2 is Newton degenerate at infinity. Moreover, if Γ+ = Γ+(f), it canbe checked using the program Gérmenes [56] that Γ+ is not simplicial. However, by astraighforward computation it follows that the function fΓ+

is Newton non-degenerate atinfinity and

µ(A(f)) = 52 < 54 = µ(A(fΓ+)).

Then f is Newton degenerate at infinity, by (3.34). On the other hand, the polynomial map∇f is Newton non-degenerate at infinity. The component functions of ∇f are given by

∂f

∂x1= 2x1 + x3

2 − 4x1x22 + 3x2

1x2

∂f

∂x2= 2x2 + x3

1 − 4x21x2 + 3x1x

22 + 3x2

2x3

∂f

∂x3= 4x3

3 + x32.

Then Γ+(∇f) = Γ+(x31 +x3

2 +x33) and ∇f is Newton non-degenerate at infinity, since µ∞(f) =

27 = 3!V3(Γ+(∇f)). However, if I = 2, then we have (∂f/∂x3)I = x32 and ∂fI/∂x3 = 0,

hence f does not satisfy condition (3.35).

Given a Newton polyhedron at infinity Γ+ ⊆ Rn>0, we define

O(Γ+) =f ∈ C[x1, . . . , xn] : Γ+(f) = Γ+ and (∇f)−1(0) is finite

.

Then, when O(Γ+) 6= ∅, we can consider the number

δ(Γ+) = maxn!Vn(Γ+(∇f)) : f ∈ O(Γ+).

by Theorem 2.2.8, we have that

ν(Γ+) = maxµ∞(f) : f ∈ O(Γ+).

Therefore, the inequality ν(Γ+) 6 δ(Γ+) holds, by Corollary 3.2.4 and these number aredifferent in general. The next example shows this fact.

Example 3.4.5. Let us consider the following set of Z2, given by

A = 3e1, 3e2, (3, 3).


Let Γ+ = Γ+(A). A direct computation shows that ν(Γ+) = 13. Let f be the polynomialgiven by f = x3

1 + x32 + x3

1x2 + x31x

32 + x1x

32. Then, it is immediate to check that f ∈ O(Γ+)

and δ(Γ+) = 2!V2(Γ+(∇f)) = 17.

We observe that the maximum δ(Γ+) is attained at some function f ∈ C[x1, . . . , xn] suchthat supp(f) = Γ+ ∩ Zn>0.

Given a convenient Newton polyhedron at infinity Γ+ ⊆ Rn>0, we define

K(Γ+) =f ∈ O(Γ+) : f is Newton non-degenerate at infinity

.

B(Γ+) =f ∈ O(Γ+) : ∇f is Newton non-degenerate at infinity

.

Proposition 3.4.6. Let Γ+ ⊆ Rn>0 be a convenient Newton polyhedron at infinity. Then

K(Γ+) ⊆ B(Γ+) if and only if ν(Γ+) = δ(Γ+).

Proof. Suppose that ν(Γ+) = δ(Γ+) and let f ∈ K(Γ+). Then we obtain the followinginequalities:

ν(Γ+) = µ∞(f) 6 n!Vn(Γ+(∇f)) 6 δ(Γ+).

Since we are assuming that ν(Γ+) = δ(Γ+), the above relation implies that µ∞(f) is equalto n!Vn(Γ+(∇f)) and then the polynomial map ∇f is Newton non-degenerate, by Corollary3.2.4.

Suppose that K(Γ+) ⊆ B(Γ+) and let f ∈ O(Γ+) such that

δ(Γ+) = n!Vn(Γ+(∇f)).

The Newton non-degeneracy condition for functions is generic in the space of polynomialfunctions g such that supp(g) ⊆ Γ+ (see [43, Theorem 6.1]). Thus, modifying the coefficientsof f , we can suppose that f is Newton non-degenerate. In particular, the polynomial map∇f is Newton non-degenerate, since we are assuming K(Γ+) ⊆ B(Γ+). Therefore

ν(Γ+) = µ∞(f) = n!Vn(Γ+(∇f)) = δ(Γ+).

It is clear that K(Γ+) 6= ∅ for any convenient Newton polyhedron at infinity Γ+. On theother hand, the family B(Γ+) is not always non-empty, as the following proposition shows.

Proposition 3.4.7. Let Γ+ ⊆ Rn>0 be a convenient Newton polyhedron. Then B(Γ+) 6= ∅

if and only if ∇g is Newton non-degenerate at infinity, for any function g ∈ C[x1, . . . , xn]such that supp(g) = ϑ(Γ+) and g is Newton non-degenerate at infinity. In particular, if Γ+

is simplicial, then B(Γ+) 6= ∅ if and only if fΓ+∈ B(Γ+).


Proof. The if part is obvious. Let us see the converse. Let f ∈ O(Γ+) be such that thepolynomial map ∇f is Newton non-degenerate at infinity. Let g ∈ C[x1, . . . , xn] such thatsupp(g) = ϑ(Γ+) and g is Newton non-degenerate at infinity. In particular, we have theinequality

Vn(Γ+(∇g)) 6 Vn(Γ+(∇f)).

Since the polynomial map ∇f and the polynomial g are Newton non-degenerate at infinity,we deduce that

ν(Γ+) > µ∞(f) = n!Vn(Γ+(∇f)) > n!Vn(Γ+(∇g)) > µ∞(g) = ν(Γ+).

Hence, we have µ∞(g) = µ(∇g) = n!Vn(Γ+(∇g)). Then the polynomial map ∇g is Newtonnon-degenerate at infinity by Corollary 3.2.4.

Example 3.4.8. Let g ∈ C[x1, x2] be the function given by g = x92 + x2x

71 + x4

1 and letΓ+ = Γ+(g). It is clear that g = fΓ+

. It is immediate to see that Γ+(∇g) = Γ+(x71 + x8

2) and∇g = (4x3

1 + 7x61x2, x

71 + 9x8

2) is Newton degenerate at infinity. Hence there is no polynomialfunction f ∈ On(Γ+) such that ∇f is Newton non-degenerate at infinity, by the previousresult. This example shows that the Newton non-degeneracy at infinity of the gradient mapis not a generic condition in C[Γ+].

3.5 Homogeneous Newton polyhedra

Motivated by the characterization of monomial ideals in C[x1, . . . , xn], whose integralclosure admits a reduction generated by homogeneous polynomials proven by Bivià-Ausinain [10]. In this section we study a class of global Newton polyhedra that called homogeneous.

We begin with some definitions and results about the determination of the number ofisolated roots of polynomial systems (see [65] and [67]).

We recall from Section 2.4 that if P = (P1, . . . , Pn) is a n-tuple of lattice of polytopesof Rn

>0, then Cn[P] denotes the set of the polynomial maps F = (F1, . . . , Fn) : Cn → Cn

such that supp(Fi) ⊆ Pi for all i = 1, . . . , n. If F−1(0) is finite, then µ(F ) 6 MV(P0) andthe conditions ‘nice’ and ‘corned’ on P imply µ(F ) = MV(P0) for a generic F ∈ Cn[P] (seeTheorem 2.4.4).

In order to develop this section we state some notation. Let P = (P1, . . . , Pn) and Q =(Q1, . . . , Qn) be n-tuples of polytopes in Rn

>0, write P ⊆ Q to denote that Pi ⊆ Qi forall i = 1, . . . , n. We also denote the n-tuples of subsets (P1 ∩ Q1, . . . , Pn ∩ Qn) and (Q1 \


P1, . . . , Qn \ Pn) by P ∩ Q and Q \ P respectively. If I ⊆ 1, . . . , n, I 6= ∅, then we setPI = (P1 ∩ Rn

I , . . . , Pn ∩ RnI ).

Next we recall a particular definition introduced in [67, p. 120].

Definition 3.5.1. Let P and Q be a n-tuples of polytopes in Rn>0 such that Q is nice and

cornered. We say that P counts Q when

(i) P ⊆ Q;

(ii) P is nice;

(iii) for any F ∈ Cn[Q \P], the map F + F′ has a finite zero set and µ(F + F

′) = MV(Q0)for a generic F ′ ∈ Cn[P].

In particular, if P counts Q, then µ(F ) = MV(Q0) for a generic F ∈ Cn[P] and thereforeMV(P0) = MV(Q0).

Definition 3.5.2. Let P = (P1, . . . , Pn) be an n-tuple of polytopes in Rn. The support of Pis defined as the set of indices i ∈ 1, . . . , n such that Pi 6= ∅. We denote this set by supp(P).Let J ⊆ 1, . . . , n. Then J is said to be essential for P when the following conditions hold

(i) J ⊆ supp(P);

(ii) dim(∑j∈J Pj) = |J | − 1;

(iii) for all nonempty proper subset J ′ ⊂ J , we have that dim(∑j∈J Pj) > |J′|.

Next we state a particular case of [67, Theorem 7] that we need for our purposes (weremark that this theorem is stated for polynomials with coefficients in any algebraicallyclosed field). Given two n-tuples of polytopes P and Q of Rn such that P ⊆ Q, this resultgives a purely combinatorial characterization of when P counts Q.

Theorem 3.5.3 ([67, p. 127]). Let P and Q be n-tuples of lattice polytopes contained in Rn>0

such that P ⊆ Q. Let us suppose that Q is nice and cornered and MV(Q0) > 0. Then Pcounts Q if and only if supp(P∩Qw) contains an essential subset for Qw for all w ∈ Rn\Rn

>0.

Let Γ+ ⊆ Rn>0 be a convenient global Newton polyhedron at infinity. Let us fix a subset

I ⊆ 1, . . . , n, I 6= ∅. We denote by ΓI+ the intersection Γ+ ∩ Rn

I and we define deg(ΓI+) =

max|k| : k ∈ ΓI+, where |k| denotes the sum of the coordinates of k, for any k ∈ Rn.

Let us define

bi(Γ+) = min

deg(ΓI+) : I ⊆ 1, . . . , n, |I| = n− i+ 1

, (3.38)

for all i ∈ 1, . . . , n. It is immediate to see that b1(Γ+) > · · · > bn(Γ+).


Definition 3.5.4. Let Γ+ be a Newton polyhedron at infinity. We say that Γ+ is homogeneousif there exist a polynomial map G = (G1, . . . , Gn) : Cn → Cn such that G Newton non-degenerate at infinity, Γ+ = Γ+(G) and Gi is a homogeneous polynomial of degree bi(Γ+), forall i = 1, . . . , n.

Theorem 3.5.5. Let Γ+ be a convenient Newton polyhedron at infinity with vertices in Zn>0.Then

n!Vn(Γ+) > b1(Γ+) · · · bn(Γ+) (3.39)

and equality holds if and only if there exists λ ∈ Z>1 and polynomial map G = (G1, . . . , Gn) :Cn → Cn such that G is Newton non-degenerate at infinity, Gi is a homogeneous polyomialof degree λbi(Γ+), for all i = 1, . . . , n and Γ+(G) = λΓ+.

Proof. Let bi = bi(Γ+), for all i = 1, . . . , n. Let us denote by Di the convex hull in Rn of theset

k ∈ ΓI+ ∩ Zn>1 : |k| = bi, I ⊆ 1, . . . , n, |I| = n− i+ 1

for all i = 1, . . . , n. By the definition of bi, we have bi = max|k| : k ∈ Γ+ ∩ RnI for some

I ⊆ 1, . . . , n such that |I| = n − i + 1, for all i = 1, . . . , n. In particular, Di 6= ∅ for alli = 1, . . . , n, since Γ+ is convenient Newton polyhedron with vertices in Zn>0.

Let us denote the n-tuple of polytopes (D1, . . . , Dn) by D. If α ∈ Z>1, let ∆(α) denotethe convex hull in Rn of the set k ∈ Zn>0 : |k| = α and ∆ denote the n-tuple of polytopes(∆(b1), . . . ,∆(bn)). It is clear that ∆ is nice and cornered and MV(∆0) = b1 · · · bn > 0.Clearly, we have that D ⊆ ∆. We claim that there exist λ ∈ Z>1, such that λD counts λ∆.To see this, we will apply Theorem 3.5.3.

Let us fix a vector w = (w1, . . . , wn) ∈ Rn \ Rn>0 and let w0 = minw1, . . . , wn. Let

Iw denote the set of indices i : wi = w0. It is immediate that `(w,∆(bj)) = bjw0 and∆(bj)w = ∆(bj) ∩ Rn

Iw for all w ∈ Rn \ Rn>0 and for all j = 1, . . . , n. Then ∆w = ∆Iw for all

w ∈ Rn \ Rn>0. In particular, we have the equality

∆w : w ∈ Rn \ Rn

>0

=∆I : I ⊆ 1, . . . , n, I 6= ∅

.

Fix a subset I ⊆ 1, . . . , n, I 6= ∅. Let α = |I| and consider the set of indices JI =n+ 1−α, . . . , n. Let us show that there exits λ ∈ Z>1, such that JI ⊆ supp((λD)∩ (λ∆)I)and JI is an essential set for (λ∆)I.

If i ∈ JI, then α > n − i + 1 and thus deg(ΓI+) > deg(ΓI′

+) > bi, for all I′ ⊆ I such that|I′| = n − i + 1. In particular, if I′ ⊆ I is any subset such that |I′| = n − i + 1, there exist


some k ∈ ΓI′+ ∩Qn

>0 ⊆ ΓI+ ∩Qn

>0 such that |k| = bi. Let λI ∈ Z>1 such that λIk ∈ Zn>1. Then(λIDi) ∩ ∆(λIbi)I 6= ∅, for all i ∈ JI. That is, we have that JI ⊆ supp((λID) ∩ (λI∆)I).Then taking λ as the least common multiple of λI where I ⊆ 1, . . . , n, we obtain thatJI ⊆ supp((λD) ∩ (λ∆)I), for all I ⊆ 1, . . . , n, I 6= ∅.

We observe that dim(∆(b))I = |I| − 1 for all b ∈ Z>1. Moreover ∑j∈JI(∆(bj))I =(∆(∑j∈JI bj))I. In particular, we have dim∑

j∈JI ∆(λbj)I = |I| − 1. Then we observe thatJI satisfies conditions (2) and (3) of the definition of essential subset for (λ∆)I. Thus, wededuce that λD counts λ∆, by Theorem 3.5.3.

In particular, there exist homogeneous polynomialsGi ∈ C[Di], i = 1, . . . , n, such that, theG = (G1, . . . , Gn) : Cn → Cn, G−1(0) is finite and µ(G) = MV((λ∆)0) = λnb1 · · · bn. SinceGi is homogeneous, for all i = 1, . . . , n, and G−1(0) is finite, we conclude that G−1(0) = 0.It is easy to see that Γ+(G) ⊆ λΓ+. This implies that λnb1 · · · bn = µ(G) 6 n!Vn(λΓ+(G)) 6n!Vn(λΓ+) and then inequality (3.39) follows. Moreover, equality (3.39) holds if and only ifG is Newton non-degenerate and Γ+(G) = λΓ+, by Corollary 3.2.4.

Remark 3.5.6. Let us consider the following property on Γ+: for all i ∈ 1, . . . , n and allI ⊆ 1, . . . , n such that |I| = n− i + 1 there exists some k ∈ ΓI

+ such that |k| = bi(Γ+). Acareful revision of the proof of Theorem 3.5.5 reveals that if Γ+ satisfies this property, thenwe can take λ = 1 in the statement of Theorem 3.5.5.

Corollary 3.5.7. Let Γ+ ⊆ Rn>0 be a convenient Newton polyhedron at infinity with vertices

in Zn>0. Suppose that Γ+ is homogeneous. Then

n!Vn(Γ+) = b1(Γ+) · · · bn(Γ+).

Proof. By definition there exist a Newton non-degenerate at infinity polynomial map G =(G1, . . . , Gn) : Cn → Cn satisfying Definition 3.5.4. Then by Theorem 2.4.4 and Corollary2.4.5, we deduce that n!Vn(Γ+) 6 b1(Γ+) · · · bn(Γ+) and by the above theorem we have theequality.

Example 3.5.8. Let us consider the following subset of Z3, given by

A = e1, e2, e3, (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1)

Let Γ+ = Γ+(A). Then a direct computation shows that b1(Γ+) = 3, b2(Γ+) = 2 andb3(Γ+) = 1. We claim that Γ+ is homogeneous. Indeed, let us consider the polynomial map


G = (G1, G2, G3) : C3 → C3, whose components functions are given by

G1(x1, x2, x3) = x1 + x2 + x3,

G2(x1, x2, x3) = x1x2 + x2x3 + x1x3,

G3(x1, x2, x3) = x1x2x3.

It is immediate to check thatG is Newton non-degenerate at infinity, eachGi is a homogeneouspolynomial of degree bi(Γ+), for all i = 1, 2, 3, and Γ+ = Γ+(G). Then, by definition, wehave that Γ+ is a homogeneous global Newton polyhedron. Let us remark that 3!V3(Γ+) =b1(Γ+)b2(Γ+)b3(Γ+).

The following example shows a global Newton polyhedron at infinity that is not homoge-neous.

Example 3.5.9. Let us consider the following subset of Z2>0 given by

A = 2e1, 3e2, (2, 2)

Let Γ+ = Γ+(A). A direct computation shows that b1(Γ+) = 4, b2(Γ+) = 2 and Γ+ satisfiesthe strict inequality of (3.39). In particular, Γ+ is not homogeneous, by Theorem 3.5.5 andRemark 3.5.6.


CHAPTER 4

Łojasiewicz exponent and adapted maps toNewton polyhedra

Contents4.1 Maps adapted to Newton polyhedra . . . . . . . . . . . . . . . . 74

4.2 Special monomials and adapted maps . . . . . . . . . . . . . . . . 81

4.3 Estimation of Łojasiewicz exponents at infinity . . . . . . . . . . 88

4.4 Łojasiewicz exponent at infinity of pre-weighted homogeneousmaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5 Index of polynomial maps . . . . . . . . . . . . . . . . . . . . . . . 94

Let us fix coordinates x1, . . . , xn in Kn. In this chapter we study the general problem ofdetermining the set of special monomials with respect to a given polynomial map F : Kn →Kp, where K = R or C. We obtain an approximation to the set of special monomials withrespect to a polynomial map in terms non-degeneracy conditions and Newton polyhedra atinfinity. The corresponding result (Theorem 4.2.4) gives rise to an estimation from below tothe Łojasiewicz exponent of a polynomial map Kn → Kp with compact zero set (Corollary4.3.3).

In order to achieve our objectives we introduce the notion of strongly non-degenerate mapwith respect to a fixed Newton polyhedron. This is basically a generalization of the notionof pre-weighted homogeneous map.

Based on our study of Łojasiewicz exponents at infinity we also derive a result about theinvariance of the index of real polynomial maps (Theorem 4.5.1).

71

72 Chapter 4. Łojasiewicz exponent and adapted maps to Newton polyhedra

4.1 Maps adapted to Newton polyhedra

Let us fix a convenient global Newton polyhedron Γ+ ⊆ Rn. In this section we exposea condition on a given polynomial map F : Kn → Kp that allows us to obtain informationabout S0(F ) and L∞(F ) in terms of Γ+. First we recall some definitions given in Section 2.1.

Definition 4.1.1. Let h ∈ K[x1, . . . , xn]. Let us suppose that h is written as h = ∑k akx

k.If w ∈ Rn \ 0, then `(w, h) and ∆(w, h) are defined as

`(w, h) = min〈w, k〉 : k ∈ supp(h).

∆(w, h) = k ∈ supp(h) : 〈w, k〉 = `(w;h).

We define the principal part of h with respect to w, denoted by pw(h), as the sum of thoseterms akxk such that 〈k, w〉 = `(w, h). We observe that if h denotes a monomial xk thenpw(h) = h, for any w ∈ Rn \ 0. If F = (F1, . . . , Fp) : Kn → Kp is a polynomial map, thenwe denote the map (pw(F1), . . . , pw(Fp)) : Kn → Kp by pw(F ).

Example 4.1.2. Let h ∈ K[x1, x2] be a polynomial given by h(x1, x2) = x31 + x4

1x42 − 4x2

1x32.

Then supp(h) = (3, 0), (4, 4), (2, 3). Let w = (3,−1), then a direct computation shows that`(w, h) = 3 and this minimum is attained only at the point (2, 3) ∈ supp(h). Then ∆(w, h) =(2, 3) and pw(h) = −4x2

1x32. Let us remark that `(w, Γ+((h)) = 0 and ∆(w, Γ+((h)) =

(0, 0). In general it is immediate to see that, if g ∈ K[x1, . . . , xn] and v ∈ R \ 0 then`(v, Γ+(g)) 6 `(v, g) and equality holds if and only if `(v, g) 6 0.

Example 4.1.3. Let F : K2 → K3 be the polynomial map where the component functionsare given by

F1(x1, x2) = x31 − 3x2

1x32 + x4

2,

F2(x1, x2) = x21 − x3

1x72 + 4x2

1x42,

F3(x1, x2) = 4x1x32 + 7x2

1x62.

Let w = (3,−1), then we have that `(w,F1) = −4, `(w,F2) = 2, `(w,F3) = 0, ∆(w,F1) =(0, 4), ∆(w,F2) = (3, 7), (2, 4) and ∆(w,F3) = (1, 3), (2, 6). Therefore, it is easy toobserve that pw(F ) is given by pw(F )(x1, x2) = (x4

2,−x31x

72 + 4x2

1x42, 4x1x

32 + 7x2

1x62).

If J ⊆ F(Γ+), then we denote by ∆J(h) the intersection ∩w∈J∆(w, h). We define theprincipal part of h with respect to J , which we will denote by pJ(h), as the sum of all termsakx

k such that k ∈ ∆J(h). If ∆J(h) = ∅, then we set pJ(h) = 0.

Chapter 4. Łojasiewicz exponent and adapted maps to Newton polyhedra 73

We denote by |A| the cardinal of a given finite set A. If ∆ is a face of Γ+, then we denote byJ(∆) the set of those subsets J ⊆ F(Γ+) such that ∆ = ∩w∈J∆(w, Γ+) and dim ∆ = n− |J |.Then we observe that J(∆) is formed by all subsets J ⊆ F(Γ+) that minimally satisfy thecondition ∆ = ∩w∈J∆(w, Γ+). In particular, if ∆ is a vertex of Γ+ then |J | = n, for allJ ∈ J(∆).

Example 4.1.4. Let Γ+ be the Newton polyhedron shown in the following figure,

e 3

-e3

-e1

2

(-2,-2,-1)

-e2

1e

(2,2,0)

x1

x2

x3

(2,0,0)

(2,0,2)

(0,0,2) (0,2,2)

(2,1,2)

(1,2,2)

e

(0,2,0)

Then, we observe that Γ+ is convenient and F(Γ+) = e1, e2, e3,−e1,−e2,−e3, w, wherew = (−2,−2,−1). We shall give some examples about the above definitions.

(i) Let h ∈ K[x1, x2, x3] be the polynomial given by h(x1, x2, x3) = x21 + x2

2 + 2x21x

22x

33.

If J = −e1,−e2,−e3, then it is easy to see that ∆J(h) = (2, 2, 3) and pJ(h) =2x2

1x22x

33, however if J = e2,−e3, then ∆J(h) = ∅ and by definition pJ(h) = 0.

(ii) Let us consider ∆1 the vertex of Γ+ given by (2, 2, 0) and the 1-dimensional face ∆2

of Γ+, given by the segment determined by the points (2, 0, 2) and (2, 1, 2). Then

J(∆1) = −e1, e3, w, −e1,−e2, w, −e2, e3, w, −e1,−e2, e3 .

J(∆2) = −e1,−e3 .

Definition 4.1.5. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. We say that F isadapted to Γ+ when for any face ∆ of Γ+ such that 0 /∈ ∆ and for all J ∈ J(∆) we have

x ∈ Kn : pJ(F1)(x) = · · · = pJ(Fp)(x) = 0⊆x ∈ Kn : x1 · · · xn = 0

.

We also refer to the above inclusion as the condition (CF,J). We will denote the map(pJ(F1), . . . , pJ(Fp)) by pJ(F ).


The previous definition is motivated by the notion of pre-weighted homogeneous map andthe Newton non-degeneracy condition on germs of analytic functions (Kn, 0) → K studiedby Kouchnirenko [43] and Yoshinaga [79]. We observe that under the conditions of theabove definition, if F satisfies the (CF,J) condition, for a given J ⊆ F(Γ+), then not all thepolynomials pJ(Fi) are identically zero. In particular, there exists some i ∈ 1, . . . , p suchthat ∩w∈J∆(w, Γ+(Fi)) is a face of Γ+(Fi).

Remark 4.1.6. (i) In the particular, when n = 2. In order to see that if a polynomialmap F : K2 → Rp is adapted to a convenient Newton polyhedron Γ+ ⊆ R2, if sufficientto check if F satisfies the condition (CF,J) for all J ⊆ F(Γ+) such that ∩w∈J∆(w, Γ+)is a face of Γ+ not containing the origin, since J(∆) = J for all face ∆ of Γ+ suchthat 0 /∈ ∆, where J is a subset of F(Γ+).

(ii) Let us consider a polynomial map F : Kn → Kp such that some of the componentfunctions of F is a monomial xk, for some k ∈ Zn>0, k 6= 0. Since pJ(xk) = xk, for anyJ ∈ F(Γ+), then F is automatically adapted to Γ+. This fact suggests that we needto strengthen the above definition in order to obtain a sufficiently restrictive class ofpolynomials F : Kn → Kp for which it is possible to obtain a lower bound for L∞(F ).

Let I ⊆ 1, . . . , n, I 6= ∅. Define KnI = x ∈ Kn : xi = 0, for all i /∈ I and we denote by πI

the natural projection Rn → RnI .

Let h ∈ K[x1, . . . , xn] and let us suppose that h is written as h = ∑k akx

k. Then wedenote by hI the sum of all terms akxk such that k ∈ supp(h) ∩ Rn

I . If supp(h) ∩ RnI = ∅,

then we set hI = 0. If F = (F1, . . . , Fp) : Kn → Kp is a polynomial map then we defineF I = (F I

1 , . . . , FIp ) : Kn

I → Kp. Let us denote by ΓI+ the projection πI(Γ+ ∩ RI

n). We remarkthat in general ΓI

+ is not equal to πI(Γ+).We will strengthen the notion of adapted maps in the next definition. Let w ∈ F(Γ+) and

let h ∈ K[x1, . . . , xn]. Then define

`∗(w, h) =`(w, h) if `(w, Γ+) < 0,0 if `(w, Γ+) = 0.

Let us suppose that h is written as h = ∑k akx

k. If J ⊆ F(Γ+), then we denote by p∗J(h) thesum of all terms akxk such that 〈k, w〉 = `∗(w, h), for all w ∈ J . If the set of such terms akxk

is empty, then we set p∗J(h) = 0.Let us observe that, since Γ+ is convenient, then F(Γ+) = F0(Γ+)∪e1, . . . , en. Therefore,

if w ∈ F(Γ+), then the condition `(w, Γ+) = 0 is equivalent to saying that w is equal to some


vector ei. Thus if J ∩ e1, . . . , en = ∅, then pJ(h) = p∗J(h). If J ∩ e1, . . . , en 6= ∅, then

p∗J(h) =

pJ(h) if `(w, h) = 0 for all w ∈ J ∩ e1, . . . , en0 if otherwise. (4.1)

Definition 4.1.7. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. We say that F isstrongly adapted to Γ+ when for any face ∆ of Γ+ such that 0 /∈ ∆ and for all J ∈ J(∆) wehave

x ∈ Kn : p∗J(F1)(x) = · · · = p∗J(Fp)(x) = 0⊆x ∈ Kn : x1 · · · xn = 0

. (4.2)

We also refer to the above inclusion as the condition (C∗F,J).

As an immediate consequence of the previous definition, if F is strongly adapted to Γ+, thenF is adapted to Γ+. In general the converse does not hold, as the following example shows.

Example 4.1.8. Let us consider the polynomial map F = (F1, F2) : C2 → C2, whosecomponents are given by

F1(x1, x2) = x31 + x7

1x62 + x4

2,

F2(x1, x2) = x61x

52.

Let g be the function defined by g(x1, x2) = x1 + x2 + x1x2 and let Γ+ = Γ+(g). Then weobserve that Γ+ is convenient and F(Γ+) = e1, e2,−e1,−e2. Then F is adapted to Γ+,since it contains a monomial, but it is not strongly adapted, because if we consider the face∆ = (0, 1) then J(∆) = e1,−e2. It is straightforward to check that p∗J(Fi) = 0, for alli = 1, 2. Therefore, by definition F is not strongly adapted to Γ+.

Proposition 4.1.9. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map such that Fj isconvenient, for all j = 1, . . . , p. Then the following conditions are equivalent:

(i) F is adapted to Γ+.

(ii) F is strongly adapted to Γ+.

Proof. It follows as an immediate application of (4.1) and Definition 4.1.7.

The following definition is concerned only with polynomial maps. That is, it is not appliedto pairs (F, Γ+) formed by a polynomial map F and a Newton polyhedron Γ+ (see Definitions4.1.5 and 4.1.7). Thus, once we fix coordinates in Kn, it can be considered as an intrinsicproperty of polynomial maps Kn → Kp.


Definition 4.1.10. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. We say that Fis non-degenerate when for all w ∈ Rn

0 we havex ∈ Kn : pw(F1)(x) = · · · = pw(Fp)(x) = 0

⊆x ∈ Kn : x1 · · · xn = 0

.

We will refer to the above inclusion as the condition (CF,w).

Under the above conditions we observe that, given a vector w ∈ Rn, the (CF,w) conditionapplied on F depends only on the family of polynomials pw(F1), . . . , pw(Fp). On the otherhand, the convex hull of a finite subset of Rn has only a finite number of faces (see [31,p. 29]). Therefore, if F : Kn → Kp is any polynomial map, the number of non redundant(CF,w) conditions applied to F , where w varies in Rn, is less than or equal to the cardinalityof (∆1, . . . ,∆p) : ∆j is a face of Γ+(Fj), j = 1, . . . , p. Hence the notion of non-degeneracyis formed by a finite family of conditions.

Example 4.1.11. Let F = (F1, F2) : C2 → C2 be the map whose component functions aregiven by

F1(x1, x2) = x31 + x3

2 + x21x

62 + x3

1x42,

F2(x1, x2) = x41x

22 + 3x2

1x62 + 4x3

1x42.

If w = (−2,−1), then pw(F1) = x21x

62 + x3

1x42 and pw(F2) = x4

1x22 + 3x2

1x62 + 4x3

1x42. We observe

that the polynomials pw(F1) and pw(F2) vanish along the line x1 = −x22. Therefore F is

degenerate. On the other hand, let us consider the polynomial map G = (G1, G2) : C2 → C2

such that G1 = F1 and G2 = x41x

22 + 3x2

1x62 = F2− 4x3

1x42. Then a simple computation reveals

that G is non-degenerate.

The definition of non-degeneracy becomes more restrictive when we suppose that each Fjis convenient, j = 1, . . . , p, as see in the next result.

Proposition 4.1.12. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map such that Fj isconvenient, for all j = 1, . . . , p. Then the following conditions are equivalent:

(i) F is non-degenerate.

(ii) F is adapted to the Minkowski sum Γ+ = Γ+(F1) + · · ·+ Γ+(Fp).

Proof. Let us see (i) ⇒ (ii). Let ∆ be a face of Γ+ such that 0 /∈ ∆. Let J ∈ J(∆).In particular ∆ = ∩w∈J∆(w, Γ+) 6= ∅. If k ∈ ∆ and we write k = k1 + · · · + kp, where


kj ∈ Γ+(Fj), for all j = 1, . . . , p, then we have kj ∈ ∆(w, Γ+(Fj)), for all j = 1, . . . , p andall w ∈ J , as a consequence of Lemma 2.1.4 (i). In particular ∩w∈J∆(w, Γ+(Fj)) 6= ∅, forall j = 1, . . . , p, and then ∩w∈J∆(w, Γ+(Fj)) = ∆(∑w∈J w, Γ+(Fj)), for all j = 1, . . . , p. Letus observe that ∆(w,Fj) = ∆(w, Γ+(Fj)), for all j = 1, . . . , p and all w ∈ J , since eachpolynomial Fj is convenient. Then we obtain the equality of polynomials pJ(Fj) = pv(Fj),for all j = 1, . . . , p, where v = ∑

w∈J w. Thus condition (CF,J) is equivalent to condition(CF,w) and the result follows. Let us see (ii)⇒ (i). Let v ∈ Rn \ 0. Then there exists someJ ⊆ F(Γ+) such that ∆(v, Γ+) = ∩w∈J∆(w, Γ+) 6= ∅ and J ∈ J(∆(v, Γ+)). Then, similarly tothe proof of the other implication, we deduce that pv(Fj) = pJ(Fj), for all j = 1, . . . , p, andhence the result follows.

Let us fix a convenient global Newton polyhedron Γ+ ⊆ Rn>0. We recall that in Section

2.3, the filtrating map associated to Γ+ is denoted by φ and M is the positive constant whereφ is constant on the boundary Γ and equal to M . If w ∈ F0(Γ+), then φw : Rn

>0 → R denotesthe linear map given by φw(k) = M

`(w,Γ+)〈w, k〉. Then, φ(k), is given by φ(k) = maxφw(k) :

w ∈ F0(Γ+). If h is a polynomial in K[x1, . . . , xn], then by definition of degree of h withrespect to Γ+ (see Definition 2.3.2), we obtain the following inequalities:

M

`(w, Γ+)〈w, k〉 = φw(k) 6 φ(k) 6 νΓ+

(h), (4.3)

for all k ∈ supp(h) and for all w ∈ F(Γ+), where νΓ+(h) denotes the degree of h with respect

to Γ+.

Proposition 4.1.13. Let F = (F1, . . . , Fn) : Kn → Kn be a finite polynomial map. Let us fixa convenient global Newton polyhedron Γ+ ⊆ Rn

>0. If F is non-degenerate with respect to Γ+,then F is strongly adapted to Γ+.

Proof. Let us fix w ∈ F0(Γ+). Since F is non-degenerate with respect to Γ+ and ∆(w, Γ+) isa face of Γ+ of dimension n− 1, there exists ki ∈ C(∆(w, Γ+)) such that νΓ+

(Fi) = νΓ+(xki).

Then, by Lemma 2.3.1, we obtain that νΓ+(Fi) = φ(ki) = φw(ki) for all i = 1, . . . , n. Using

the inequality (4.3), we obtain that

M

`(w, Γ+)〈w, k〉 6 νΓ+

(Fi) = M

`(w, Γ+)〈w, ki〉, (4.4)

for all k ∈ supp(Fi). Thus, the above inequality shows that `(w,Fi) = 〈w, ki〉, and

νΓ+(Fi) = M

`(w, Γ+)`(w,Fi), for all w ∈ F0(Γ+) and for all i = 1, . . . , n. (4.5)


Let ∆ be a face of Γ+ such that 0 /∈ ∆ and let J ∈ J(∆), then we can decompose J asJ = J1 ∪ J2, where J2 = J ∩e1, . . . , en and J1 = J \ J2. Let us consider the following sets:

Ai = k ∈ supp(Fi) : k ∈ C(∆) and νΓ+(Fi) = φ(k),

BiJ = k ∈ supp(Fi) : 〈w, k〉 = `(w,Fi), for all w ∈ J1 and 〈w, k〉 = 0, for all w ∈ J2.

We claim that Ai = BiJ , for all i = 1, . . . , n. Since ∆ = (⋂w∈J1 ∆(w, Γ+))∩ (⋂w∈J2 ∆(w, Γ+)),

the claim follows as an immediate application of (4.3), (4.5) and Lemma 2.3.1.By the above affirmation, we have that in∆(Fi) = p∗J(Fi), for all i = 1, . . . , n. Therefore

the result follows.

The next example shows that the converse of the last proposition does not hold.

Example 4.1.14. Let us consider a, b,m, n ∈ Z>1, such that a 6 m and b 6 n and letF = (F1, F2) : C2 → C2 be a polynomial map, whose component functions are given by

F1(x1, x2) = xa1 + xm1 xn2 ,

F2(x1, x2) = xb2.

Let Γ+ = Γ+(F ) ⊆ R2. Then we observe that Γ+ is convenient and F(Γ+) = e1, e2, w1, w2,where w1 and w2 are the primitive vectors in Z2, such that ∆(w1, Γ+) is the face of Γ+

determined by the points (0, b) and (m,n) and ∆(w2, Γ+) is the face of Γ+ determined bythe points (a, 0) and (m,n), as is shown in the Figure 4.1. It is immediate to see that F is

w

w

1

2

(m,n)

x

x1

2

a

b

Figure 4.1: Γ+

strongly adapted to Γ+. However F is degenerate with respect to Γ+. To see this it suffices


to consider ∆ = ∆(w2, Γ+) and to observe that(x1, x2) ∈ C2 : in∆(F1)(x) = in∆(F2)(x) = 0

=

(x1, x2) ∈ C2 : F1(x1, x2) = 0

6⊆

(x1, x2) ∈ C2 : x1 · x2 = 0.

Remark 4.1.15. From Proposition 2.3.9, we have that the class of Newton non-degenerateat infinity polynomial maps is contained in the class of the polynomial maps which are non-degenerate with respect to a given global Newton polyhedron and from Proposition 4.1.13,this class is contained in the class of the strongly adapted polynomial maps. We also can seein the Remark 2.3.7 and in the example 4.1.14 that these are strictly contained.

4.2 Special monomials and adapted maps

Along this section, we denote by Γ+ a convenient Newton polyhedron at infinity containedin Rn

>0. Let us recall that F(Γ+) = F0(Γ+) ∪ e1, . . . , en, where e1, . . . , en denotes thecanonical basis of Rn and F0(Γ+) are the primitive vectors supporting some face of Γ+ ofdimension n− 1 not passing through the origin.

We need to introduce some definitions before giving results that allows us to obtain amethod to obtain an approximation of special monomials with respect to polynomials mapsadapted to Γ+.

Let a1, . . . , ar ∈ Rn such that ai 6= 0, for all i = 1, . . . , r. The set σ = R>0a1 + · · ·+R>0a

r

is called the cone spanned, or generated, by a1, . . . , ar. This is also known as the positivehull of a1, . . . , ar. If σ is minimally generated by a1, . . . , ar and ai is a primitive vector ofZn, for all i = 1, . . . , r, then we will say that a1, . . . , ar are the primitive generators of σ.The intersection of σ with a supporting hyperplane of σ is called a face of σ. We define thedimension of the cone σ = R>0a

1 + · · ·+ R>0ar, denoted by dim(σ), as the dimension of the

real vector subspace spanned by a1, . . . , ar. We say that σ is simplicial when dim(σ) = r.Let us consider the equivalence relation in Rn defined as follows. If u, v ∈ Rn, then u ∼ v

if and only if ∆(u, Γ+) = ∆(v, Γ+). Obviously the corresponding quotient space X = Rn/ ∼is bijective with the set of faces of Γ+.

If ∆ is a face of Γ+, then we denote by [∆] the closure, in the euclidean sense, of the setof vectors supporting ∆. Hence [∆] is equal to a cone R>0a

1 + · · ·+R>0ar, for some primitive

vectors a1, . . . , ar ∈ Zn. In particular, if ∆ has dimension n− 1 and ∆ = ∆(w, Γ+), for somew ∈ Zn \ 0, then [∆] = R>0w. It is immediate to see that dim[∆] = n − dim ∆, for eachface ∆ of Γ+.


Given a cone σ = R>0a1 + · · · + RΓ+0a

r ⊆ Rn, by Caratheodory’s theorem (see [29, p.139]), we have that σ can be expressed as the union of cones σ1, . . . , σm of Rn such that

(i) σi ∩ σj is a face of σi and of σj , for all i, j ∈ 1, . . . ,m,

(ii) each cone σi is written as R>0ai1 + · · · + R>0a

is , where 1 6 i1 < · · · < is 6 r andai1 , . . . , ais is linearly independent.

This fact also follows from [29, p. 147, Theorem 1.12]. Then we can decompose σ as the unionof simplicial cones with generators contained in a1, . . . , ar. We call such a decompositiona simplicial subdivision of σ. Let us fix a simplicial subdivision of each n-dimensional cone[∆], where ∆ denotes a vertex of Γ+. Then we denote by Σ(n) the set of simplicial conesof dimension n arising from the fixed simplicial subdivisions of [∆], for any vertex ∆ of Γ+.Attached to each σ ∈ Σn, we will consider the primitive vectors a1(σ), . . . , an(σ) arising fromthe simplicial structure of σ. We choose an order of these vectors that will be fixed throughoutthis section. Moreover, we write ai(σ) = (ai1(σ), . . . , ain(σ)), for i = 1, . . . , n.

For each σ ∈ Σn, let us consider a copy of Kn which we will denote by Kn(σ). We willwrite the elements of Kn(σ) as yσ = (yσ,1, . . . , yσ,n). Let Wσ = yσ ∈ Kn(σ) : 0 < |yσ,j| 61, for all j = 1, . . . , n, for all σ ∈ Σ(n), and let V = x ∈ (K \ 0)n : maxi |xi| > 1. Let usconsider, for each σ ∈ Σ(n), the monomial map πσ : Wσ → (K \ 0)n given by

πσ(yσ,1, . . . , yσ,n) = (ya11(σ)σ,1 · · · yan1 (σ)

σ,n , · · · , ya1n(σ)σ,1 · · · yann(σ)

σ,n ),

where we suppose that a1(σ), . . . , an(σ) are the primitive generators of σ.

Lemma 4.2.1. Let W be the disjoint union of all sets Wσ, where σ ∈ Σ(n). Let π : W →(K \ 0)n be the map defined by π(yσ) = πσ(yσ), for all yσ ∈ Wσ, σ ∈ Σ(n). Then therestriction π|π−1(V )

: π−1(V )→ V is surjective.

Proof. We will develop the proof in the case K = C. The case K = R is analogous. Letx = (x1, . . . , xn) ∈ V . Let us write xj as xj = rje

2παj i, where rj ∈ [0,+∞[, αj ∈ [0, 1[, for allj = 1, . . . , n and i =

√−1. Let us define the vector v(x) = (− log r1, . . . ,− log rn) ∈ Rn.

Since Σn is a partition of Rn, there exists a n-dimensional cone σ ∈ Σn such that v(x) ∈ σ.Let a1(σ), . . . , an(σ) be a linearly independent set of vectors of Rn such that σ = R>0a

1(σ)+· · ·+ R>0a

n(σ). Thus, there are β1, . . . , βn > 0 such that v(x) = β1a1(σ) + · · ·+ βna

n(σ).Let rσ,j = e−βj 6 1, for all j = 1, . . . , n. We observe that

ra1j (σ)σ,1 · · · ra

nj (σ)σ,n = e−a

1j (σ)β1−···−anj (σ)βn = rj, (4.6)


for all j = 1, . . . , n. Then we consider yσ = (yσ,1, . . . , yσ,n) ∈ Wσ, where yσ,j = rσ,je2πθσ,j i,

for some θσ,j ∈ [0, 1[, j = 1, . . . , n. Using 4.6 we observe that πσ(yσ) = x if and only if thevector θσ = (θσ,1, . . . , θσ,n) verifies that

a1j(σ)θσ,1 + · · ·+ anj (σ)θσ,n ≡ αj mod Z,

for all j = 1, . . . , n. But this system has solution, because a1(σ), . . . , an(σ) are linearlyindependent over R. Hence, there exists θσ such that πσ(yσ) = x. Moreover, we observe that0 < rσ,j 6 1, for all j = 1, . . . , n, then yσ ∈ Wσ and hence π is surjective.

Let σ ∈ Σ(n) and let xk = xk11 · · ·xknn a monomial, we observe that

xk πσ(yσ) = y〈k,a1(σ)〉σ,1 . . . y〈k,a

n(σ)〉σ,n . (4.7)

In order to simplify the notation, if w ∈ F(Γ+) then in this section we will denote thenumber `(w, Γ+) only by `(w). Let us fix a cone σ ∈ Σ(n) and let a1(σ), . . . , an(σ) be theprimitive generators of σ. Since a1(σ), . . . , an(σ) ⊆ F(Γ+), some vector aj(σ) can coincidewith some vector of the canonical basis. Then, we can assume that

j : `(aj(σ)) < 0 = 1, . . . , r, (4.8)

j : `(aj(σ)) = 0 = r + 1, . . . , r + s, (4.9)

for some integers r, s > 0 such that r + s = n. Hence, if s > 1, there exist indices 1 6 i1 <

· · · < is 6 n such that ar+j(σ) = eij , for all j = 1, . . . , s.Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. Let us fix an index i ∈ 1, . . . , p.Let us suppose that Fi is written as Fi = ∑

k akxk. Let us define

Zi = k ∈ supp(Fi) : 〈k, aj(σ)〉 = `∗(aj(σ), Fi), for some j ∈ 1, . . . , n. (4.10)

ZJ = k ∈ supp(Fi) : 〈k, aj(σ)〉 = `∗(aj(σ), Fi), if and only if j ∈ J. (4.11)

for all J ∈ 1, . . . , n. It is clear that, if J1, J2 ∈ 1, . . . , n and ZJ1 ∩ ZJ2 6= ∅ then J1 = J2.Therefore Zi is the disjoint union ∪JZJ , where J varies in the set of non-empty subsets of


1, . . . , n. Hence we have that

Fi πσ(yσ) =( ∑k∈Zi

akxk +

∑k/∈Zi

akxk

) πσ(yσ)

=∑

J⊆1,...,n

∑k∈ZJ

aky`(a1(σ),Fi)σ,1 . . . y`(a

r(σ),Fi)σ,r

( ∏j /∈J

`(aj(σ))<0

y〈k,aj(σ)〉−`(aj(σ),Fi)σ,j

)( ∏j /∈J

`(aj(σ))=0

ykij−rσ,j

)

+∑k/∈Zi

aky〈k,a1(σ)〉σ,1 · · · y〈k,ar(σ)〉

σ,r yki1σ,r+1 · · · ykisσ,n (4.12)

= y`(a1(σ),Fi)σ,1 . . . y`(a

r(σ),Fi)σ,r

( ∑J⊆1,...,n

∑k∈ZJ

ak

( ∏j /∈J

`(aj(σ))<0

y〈k,aj(σ)〉−`(aj(σ),Fi)σ,j

)( ∏j /∈J

`(aj(σ))=0

ykij−rσ,j

)

+∑k/∈Zi

aky〈k,a1(σ)〉−`(aj(σ),Fi)σ,1 · · · y〈k,ar(σ)〉−`(ar(σ),Fi)

σ,r yki1σ,r+1 · · · ykisσ,n

). (4.13)

We denote by F ∗σ,i the polynomial such that

Fi πσ(yσ) = y`(a1(σ),Fi)σ,1 · · · y`(ar(σ),Fi)

σ,r · F ∗σ,i(yσ), (4.14)

for all yσ ∈ Wσ. That is F ∗σ,i is the polynomial in the variables yσ,1, . . . , yσ,n given by theexpression that appears in (4.12) and (4.13) into parenthesis.

If M > 0 then we denote by VM the set x ∈ (K \ 0)n : ‖x‖ >M.

Proposition 4.2.2. Let us suppose that F : Kn → Kp is strongly adapted to Γ+. Then forall σ ∈ Σ(n) there exists a constant Mσ > 0 such that

infyσ∈π−1

σ (VMσ )supi|F ∗σ,i(yσ)| 6= 0.

Proof. Let us assume the opposite. That is, let σ such that

infyσ∈π−1

σ (VM )supi|F ∗σ,i(yσ)| = 0,

for all M > 0. Then there exists a sequence ymm>1 ⊆ Wσ such that πσ(ym)m>1 → ∞and F ∗σ,i(ym)m>1 → 0 as m→∞, for all i = 1, . . . , p.Let W σ denote the closure of Wσ, that is W σ = yσ ∈ Kn(σ) : ‖yσ‖ 6 1. Let y =(y1, . . . , yn) ∈ W σ be a limit point of the sequence ymm>1. Let J0 = j : yj = 0. We havethat J0 6= ∅, since πσ(ym)m>1 →∞. Moreover F ∗σ (y) = 0, for all i = 1, . . . , p.Let a1(σ), . . . , an(σ) be the primitive generators of σ. The condition πσ(ym)m>1 → ∞implies that there exists some j ∈ J0 such that aj(σ) has some negative component. In


particular `(aj(σ)) < 0, since Γ+ is convenient. Therefore 0 /∈ ∆(aj(σ), Γ+) for some j ∈ J0,which is to say that the face ∩j∈J0∆(aj(σ), Γ+) does not contain the origin.We observe that

F ∗σ,i(y) =∑

J⊆1,...,nJ0⊆J

∑k∈ZJ

ak

( ∏j /∈J

`(aj(σ))<0

y〈k,aj(σ)〉−`(aj(σ),Fi)j

)( ∏j /∈J

`(aj(σ))=0

ykij−rj

). (4.15)

On the other hand, given any zσ = (zσ,1, . . . , zσ,n) ∈ Wσ, we have

p∗J0(Fi) πσ(zσ) =∑

〈k,aj(σ)〉=`∗(aj(σ),Fi)for all j∈J0

ak · z〈k,a1(σ)〉

σ,1 · · · z〈k,an(σ)〉σ,n (4.16)

= z`(a1(σ),Fi)σ,1 · · · z`(ar(σ),Fi)

σ,r

∑J⊆1,...,nJ0⊆J

∑k∈ZJ

ak

( ∏j /∈J

`(aj(σ))<0

z〈k,aj(σ)〉−`(aj(σ),Fi)σ,j

)( ∏j /∈J

`(aj(σ))=0

zkij−rσ,j

)

(4.17)

Let us consider the point y = (y1, . . . , yn) defined by

yj =

yj, if j /∈ J0,

1, if j ∈ J0.

Comparing (4.15) and (4.17) we obtain

p∗J0(Fi) πσ(y) =∏j /∈J0

y`(aj(σ),Fi)σ,j F ∗σ,i(y) = 0, (4.18)

for all i = 1, . . . , p. Since ∩j∈J0∆(aj(σ), Γ+) is a face of Γ+ not containing the origin, therelation (4.18) gives a contradiction.

The next results are tools that allows us to give approximations to the set S0(F ) andobserve that the argument of their proofs are similar. As we will see in the following section,these theorems motivate a important result (see Corollary 4.3.1) that allows us to obtain anlower estimate for Łojasiewicz exponent of a polynomial map from Kn to Kp in terms of aconvenient Newton polyhedron.

Theorem 4.2.3. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. Let us suppose thatF is adapted to Γ+. Let k ∈ Zn>0 such that

〈k, w〉 > max`(w,F1), . . . , `(w,Fp), (4.19)

for all w ∈ F(Γ+). Then k ∈ S0(F ).


The proof of the above theorem will be given in the end of this section. Let us remarkthat inequality (4.19) is assumed for any w ∈ F(Γ+) in Theorem 4.2.3. We will see thatthe same conclusion holds if we assume (4.19) only for the vectors w ∈ F0(Γ+) such that0 /∈ ∆(w, Γ+) and F is strongly adapted to Γ+. This fact is shown in the following result,which is independent from Theorem 4.2.3.

Theorem 4.2.4. Let F = (F1, . . . , Fp) : Kn → Kp be a polynomial map. Let us suppose thatF is strongly adapted to Γ+. Let k ∈ Zn>0 such that

〈k, w〉 > max`(w,F1), . . . , `(w,Fp), (4.20)

for all w ∈ F0(Γ+). Then k ∈ S0(F ).

Proof. Let us fix a cone σ ∈ Σ(n). By Proposition 4.2.2 there exists positive constants Dσ

and Mσ such thatDσ < sup

i|F ∗σ,i(yσ)|,

for all yσ ∈ π−1σ (VMσ). Then, for any yσ ∈ π−1

σ (VMσ) we have the following chain of inequalities:

supi|Fi(x)| πσ(yσ) = sup

i|y`(a

1(σ),Fi)σ,1 · · · y`(ar(σ),Fi)

σ,r · F ∗i (yσ)| (4.21)

> |yσ,1|maxi `(a1(σ),Fi) · · · |yσ,r|maxi `(a1(σ),Fi) supi|F ∗σ,i(yσ)| (4.22)

> |yσ,1|〈k,a1(σ)〉 · · · |yσ,r|〈k,a

r(σ)〉 ·Dσ (4.23)

> |yσ,1|〈k,a1(σ)〉 · · · |yσ,r|〈k,a

r(σ)〉|yσ,r+1|ki1 · · · |yσ,n|kir ·Dσ (4.24)

= Dσ|xk| πσ(yσ). (4.25)

LetM = maxσ∈Σ(n) . We can assume√n 6M . Then V contains VM and by Lemma 4.2.1, we

have that πσ(yσ) : yσ ∈ π−1σ (VM), σ ∈ Σ(n) = VM . In particular, if C = (minσ∈Σ(n) Dσ)−1

we conclude that|xk| 6 C sup

i|Fi(x)|, (4.26)

for all x ∈ (Kr 0)n such that ‖x‖ > M . By continuity we obtain that (4.26) holds for allx ∈ Kn such that ‖x‖ >M .

Proof of Theorem 4.2.3. Let us modify the definitions of Zi and of ZJ , in (4.10) and (4.11)respectively, by replacing `∗(aj(σ), Fi) by `(aj(σ), Fi). Then we obtain, as in (4.14), a poly-nomial F ′σ,i ∈ K[yσ,1, . . . , yσ,n] such that

Fi πσ(yσ) = y`(a1(σ),Fi)σ,1 · · · y`(a

n(σ),Fi)σ,j F

′

σ,i(yσ),


for all yσ ∈ Wσ and all i = 1, . . . , p. Following the proof Proposition 4.2.1, we obtain that foreach σ ∈ Σ(n) there exists a constant Mσ > 0 such that

infyσ∈π−1

σ (VMσ)

supi|F ′σ,i(yσ)| 6= 0.

Hence we can reproduce the argument of the proof of Theorem 4.2.4 to obtain that k ∈S0(F ).

Example 4.2.5. Let us consider the polynomial map F : C2 → C2, where F is given byF (x1, x2) = (xa1 + xb1

2 + xm1 xn2 , x

a1 + xb2, where a1 < a, b1 < b and mb + na > ab. Let

Γ+ = Γ+(F ) ⊆ R2>0, then Γ+ is a convenient Newton polyhedron since that F is convenient.

Let us consider w1, w2 the primitive vectors in R20, such that F(Γ+) = e1, e2, w1, w2.

It is straightforward to check that F is strongly adapted to Γ+ and a direct computationshows that `(w1, F1) = `(w1, Γ+), `(w2, F1) = `(w2, Γ+), `(w1, F2) = 〈w1, (0, b1)〉 and `(w2, F2) =〈w2, (0, a1)〉. Then by Theorem 4.2.4, we have k ∈ Z2

>0 : 〈k, w1〉 > 〈w1, (0, b1)〉 and 〈k, w2〉 >〈w2, (a1, 0)〉 ⊆ S0(F ).

The Figure 4.2 shows the Newton polyhedron of Γ+(F ) and the region where the supportof a polynomial h must be contained to be special with respect to F .

w

w

1

2

(m,n)

b

x

x1

2

a

b1

1a

(m,n)

x

x1

2

aa1

b

b1

Figure 4.2:

Corollary 4.2.6. Let F = (F1, . . . , Fn) : Kn → Kn be a polynomial map. Let us suppose thatF is non-degenerate with respect to Γ+. Let φ : Rn

>0 → Rn be the Newton filtration associatedto Γ+ and let di be the degree of Fi with respect to Γ+, for all i = 1, . . . , n. Then

k ∈ Zn>0 : φ(k) 6 d0⊆ S0(F ),

where d0 = mind1, . . . , dn.


Proof. The result follows as a direct consequence of Proposition 4.1.13, Theorem 4.2.4 andrelation 4.5.

4.3 Estimation of Łojasiewicz exponents at infinity

Let us fix along this section a convenient global Newton polyhedron Γ+ ⊆ Rn and a polyno-mial map F = (F1, . . . , Fp) : Kn → Kp. Let us define L(w,F ) = max`(w,F1), . . . , `(w,Fp),for any vector w ∈ Rn.

Corollary 4.3.1. Let us suppose that F is strongly adapted to Γ+. Let k ∈ Zn>0 and θ > 0such that θ〈k, w〉 > L(w,F ), for all w ∈ F0(Γ+). Then there exist positive constants C andM such that

|xk|θ 6 C‖F (x)‖,

for all x ∈ Kn such that ‖x‖ >M .

Proof. It follows by the same argument of the proof of Theorem 4.2.4 by replacing inequalities(4.21)-(4.25) by the following inequalities:

supi|Fi(x)| πσ(yσ) = sup

i|y`(a

1(σ),Fi)σ,1 · · · y`(ar(σ),Fi)

σ,r · F ∗i (yσ)| (4.27)

> |yσ,1|maxi `(a1(σ),Fi) · · · |yσ,m|maxi `(a1(σ),Fi) supi|F ∗σ,i(yσ)| (4.28)

> |yσ,1|θ〈k,a1(σ)〉 · · · |yσ,r|θ〈k,a

r(σ)〉 ·Dσ (4.29)

> |yσ,1|θ〈k,a1(σ)〉 · · · |yσ,r|θ〈k,a

r(σ)〉|yσ,r+1|θki1 · · · |yσ,n|θkis ·Dσ (4.30)

= Dσ|xk| πσ(yσ). (4.31)

Let us remark that the condition θ > 0 is used to obtain (4.30), since we assume 0 < |yσ,i| 6 1,for all i = 1, . . . , n and all σ ∈ Σ(n).

Let us fix an index i ∈ 1, . . . , n. Then we define

Ei(F, Γ+) = θ > 0 : θwi > L(w,F ) for allw ∈ F0(Γ+). (4.32)

Let us decompose F0(Γ+) as F0(Γ+) = Ai,−∪Ai,0∪Ai,+ where Ai,− = w ∈ F0(Γ+) : wi < 0,Ai,0 = w ∈ F0(Γ+) : wi = 0 and Ai,+ = w ∈ F0(Γ+) : wi > 0. We define

ai(F, Γ+) = maxw∈Ai,+

L(w,F )wi

bi(F, Γ+) = minw∈Ai,−

L(w,F )wi

. (4.33)

In the remaining section we also denote the numbers defined above by ai and bi, respectively,for all i = 1, . . . , n. Let us remark that Ei(F, Γ+) = [0, bi], whenever Ei(F, Γ+) 6= ∅.


Lemma 4.3.2. The following conditions are equivalent:

(i) Ei(F, Γ+) 6= ∅, for all i = 1, . . . , n.

(ii) `(w,Fj) 6 0, for all w ∈ F0(Γ+) and all j = 1, . . . , p.

Proof. Let us fix an index i ∈ 1, . . . , n. By (4.32) and (4.33) we obtain that Ei(F, Γ+) 6= ∅if and only if

ai 6 bi, 0 6 bi and `(w,Fj) 6 0, for all j = 1, . . . , p and all w ∈ Ai,0. (4.34)

Let us assume (i). If w ∈ F0(Γ+), then w0 < 0. In particular w0 = wi < 0, for somei ∈ 1, . . . , n. This implies w ∈ Ai,−. Since bi > 0, we conclude L(w,F ) 6 0, and then (ii)follows. The converse is obvious using (4.34).

In particular, if Fj is convenient, for all j = 1, . . . , p then Ei(F, Γ+) 6= ∅, for all i =1, . . . , n.

Corollary 4.3.3. Let us suppose that F : Kn → Kp is a polynomial map strongly adapted toΓ+ and Ei(F, Γ+) 6= ∅, for all i = 1, . . . , n. Then

min

L(w,F )w0

: w ∈ F0(Γ+)6 L∞(F ). (4.35)

Proof. Since F is strongly adapted to Γ+ and Ei(F, Γ+) 6= ∅, for all i = 1, . . . , n, we canapply Corollary 4.3.1 to each monomial xi, i = 1, . . . , n, to obtain that there exist constantsC, M > 0 such that

|xi|bi 6 C‖F (x)‖, (4.36)

for all x ∈ Kn such that ‖x‖ > M . Let b0 = minb1, . . . , bn. We can assume M >√n. If

‖x‖ > M then√n 6 M 6 ‖x‖ 6

√nmaxi |xi|. In particular 1 6 maxi |xi| and then (4.36)

implies thatmax |xi|b0 6 C‖F (x)‖,

for all x ∈ Kn such that ‖x‖ >M . Therefore b0 6 L∞(F ). Let us observe that

b0 = mini

minw∈Ai,−

L(w,F )wi

= minw∈F0(Γ+)

L(w,F )w0

.

Hence (4.35) follows.


In particular, by Lemma 4.3.2, the conclusion of the above result follows if we replacethe condition Ei(F, Γ+) 6= ∅, for all i = 1, . . . , n, by the condition that Fj is convenient, forall j = 1, . . . , p. Let us denote the left hand side of (4.35) by b(F, Γ+). Let us remark thatb(F, Γ+) = mini bi(F, Γ+). We remark that Corollary 4.3.3 applies both to real and complexmaps and is analogous to [8, Theorem 5.9], which applies only to real polynomial maps. Ifh ∈ K[x1, . . . , xn] and h is convenient, then we denote by ri(h) the number ri(Γ+(h)), asdefined in (2.6), for all i = 1, . . . , n.

Corollary 4.3.4. Let us suppose that F is non-degenerate and Fj is convenient for all j =1, . . . , p. Then

mini,j

ri(Fj) 6 L∞(F ) 6 r0(Γ+(F )). (4.37)

Proof. The right hand side of (4.37) follows by [8, Lemma 3.3] (the proof of this result isgiven for K = R but it also applies to the case K = C). By Proposition 4.1.12, the map F isstrongly adapted to the polyhedron Γ+ = Γ+(F1) + · · ·+ Γ+(Fp). By Lemma 2.1.6 we have

ri(Fj) = minw∈Rn0 (i)

`(w,Fj)wi

for all j = 1, . . . , p, i = 1, . . . , n. Then

minr1(Fs), . . . , rn(Fs)

= min

w∈Rn0j=1,...,p

`(w,Fj)w0

6 min

L(w,F )w0

: w ∈ F0(Γ+)6 L∞(F ),

where the last inequality comes from Corollary 4.3.3.

Next, we state the theorem of Hadamard that, in conjunction with our results on the esti-mation of Łojasiewicz exponents at infinity, will lead us to formulate results on the bijectivityof polynomial maps from Rn to Rn.

Theorem 4.3.5 (Hadamard’s Theorem [76, p. 240]). Let F : Rn → Rn be a C1 map. ThenF is a homeomorphism if and only if F is a local homeomorphism and F is proper.

Corollary 4.3.6. Let F : Kn → Kn be a polynomial map such that F is a local homeomor-phism. Let us suppose that F is strongly adapted to Γ+ and Ei(F, Γ+) 6= ∅, for all i = 1, . . . , n.Then F is a homeomorphism.

Proof. The hypothesis imply that L∞(F ) exists and L∞(F ) > 0. Then F is a proper mapand therefore, by Hadamard’s theorem, F is a homeomorphism.


Remark 4.3.7. Let us observe that, in the previous result, we do not assume that eachcomponent function of F is convenient. Corollary 4.3.4 is proven in [8, Theorem 3.8] only forreal polynomial maps. We remark that, under the conditions of Corollary 4.3.4, if we assumethat F is a local homeomorphism, then the same proof of the previous result works to deducethat F is a global homeomorphism.

4.4 Łojasiewicz exponent at infinity of pre-weighted ho-mogeneous maps

In this section we expose some results concerning in the estimate of Łojasiewicz exponentof pre-weighted homogeneous maps (see Definition 2.3.5) from Kn → Kp, which is part of themotivation of our study.

Let v = (v1, . . . , vn) ∈ Zn>1 be a primitive vector. Then we recall that denote by Γv+the global Newton polyhedron given by the convex hull of v1···vn

v1e1, . . . ,

v1···vnvn

en ∪ 0.Then Γv+ has a unique face ∆ of dimension n − 1, which is contained in the hyperplanev1x1 + · · ·+ vnxn = v1 · · · vn and hence F(Γv+) = −v, e1, . . . , en and F0(Γv+) = −v. Let usrecall that, if I is a non-empty subset of 1, . . . , n, then πI : Rn → Rn

I denotes the naturalprojection.

Lemma 4.4.1. Let F : Kn → Kp be a polynomial map and let v ∈ Zn>1. Then the followingconditions are equivalent:

(i) F is pre-weighted homogeneous with respect to v.

(ii) F is strongly adapted to Γv+.

Proof. Let us see (i) ⇒ (ii). Let w = −v. Let ∆ be a face of Γv+ such that 0 /∈ ∆. If ∆ isnot contained in any coordinate subspace Rn

I , for some proper subset I ⊆ 1, . . . , n, then∆ = ∆(w, Γv+) and J(∆) = w. Hence condition (Cw,F ) follows, since pw(F ) = qv(F )and qv(F )−1(0) = 0.If ∆ ⊆ Rn

I , for some proper subset I ⊆ 1, . . . , n such that dim ∆ = |I|−1, then J(∆) = J,where J = w, ei1 , . . . , eis and i1, . . . , is = 1, . . . , n \ I, s = n − |I|. Then we observethat p∗J(F ) = pπI(w)(F I) = qπI(v)(F I). The map qπI(v)(F I) is weighted homogeneous withrespect to πI(v). Therefore the condition qv(F )−1(0) = 0 implies qπI(v)(F I)−1(0) = 0and hence pJ(F )−1(0) = 0. Thus (C∗F,J) holds and (ii) follows, by Definition 4.1.7.

Let us see (ii) ⇒ (i). Since F0(Γv+) = −v it is clear that Ei(qv(F ), Γv+) 6= ∅, for alli = 1, . . . , n (see Lemma 4.3.2). By Definition 4.1.5 we observe that F is strongly adapted


to Γ+ if and only if qw(F ) is strongly adapted to Γv+. Then we can apply Corollary 4.3.3 toqv(F ) to deduce that there exists constants C,M > 0 such that

‖x‖α 6 C‖F (x)‖,

for all x ∈ Kn such that ‖x‖ > M . In particular qv(F )−1(0) is contained in the open ballB(0;M) centered at 0 and of radius M . But this implies qv(F )−1(0) = 0 since qv(F ) isweighted homogeneous with respect to v. Thus F is pre-weighted homogeneous.

If v = (v1, . . . , vn) ∈ Rn, then we define A(v) = j : vj = maxj vj.

Proposition 4.4.2. Let F : Kn → Kp be a polynomial map. Let v ∈ Zn>1 such that F ispre-weighted homogeneous with respect to v. Then

mindv(F1), . . . , dv(Fp)maxv1, . . . , vn

6 L∞(F ). (4.38)

Let us assume that F is weighted homogeneous with respect to v and F−1(0) = 0. Leti0 ∈ 1, . . . , p such that mindv(F1), . . . , dv(Fp) = dv(Fi0). If

x ∈ Kn : Fi(x) = 0, for all i 6= i0 6⊆ x ∈ Kn : xj = 0, for all j ∈ A(v), (4.39)

then equality holds in (4.38).

Proof. By Lemma 4.4.1, the map F is strongly adapted to Γv+. Then (4.38) follows fromCorollary 4.3.3, since F0(Γv+) = v. Let us see the second part. In order to simplifythe notation, let us assume i0 = 1. Then the quotient on the left hand side of (4.38) isequal to dv(F1)/maxi vi. By (4.39) and the hypothesis F−1(0) = 0, there exists a pointa = (a1, . . . , an) ∈ Kn such that F2(a) = · · · = Fp(a) = 0, F1(a) 6= 0 and aj 6= 0, for somej ∈ A(v). In particular the curve γ : K \ 0 → Kn defined by γ(t) = (a1t

−v1 , . . . , ant−vn) is

not the zero curve. We observe that ord(γ) = −(maxi vi). Moreover, since we assume thatF is weighted homogeneous with respect to v, we have

F (γ(t)) = (t−dv(F1)F1(a), t−dv(F2)F2(a), . . . , t−dv(Fp)Fp(a)) = (t−dv(F1)F1(a), 0, . . . , 0).

This shows that F γ is not the zero curve. Let β > dv(F1)/maxi vi. Then we have

limt→0

‖F (γ(t))‖‖γ(t)‖β = lim

t→0

‖(t−dv(F1)F1(a), 0, . . . , 0)‖‖γ(t)‖β = 0, (4.40)

where the last equality follows from

ord(‖γ(t)‖β) = −(maxivi)β < −dv(F1) = ord(F (γ(t)).


In particular L∞(F ) 6 β (otherwise the limit (4.40) would be greater than or equal to somepositive constant). Therefore L∞(F ) 6 dv(F1)/maxi vi and the result follows.

Example 4.4.3. Let F = (F1, F2, F3) : C3 → C3 be the polynomial map whose componentfunctions are given by

F1(x1, x2, x3) = x51 + x3

1x32 + x5

3,

F2(x1, x2, x3) = x92 + x2

1x32x

23 − x6

3,

F3(x1, x2, x3) = x51x

32.

Let v be the primitive vector given by v = (3, 2, 3). It is easy to observe that F is weighted-homogeneous with respect to v and dv(F1) = 15, dv(F2) = 18 and dv(F3) = 21. Moreover,A(v) = 1, 3 and dv(F1) = mindv(F1), dv(F2), dv(F3). It is easy to see that

x ∈ C3 : F2(x) = F3(x) = 0 6⊆ x ∈ C3 : x1 = x3 = 0,

since (0, 1, 1) ∈ x ∈ C3 : F2(x) = F3(x) = 0 \ x ∈ C3 : x1 = x3 = 0. Then by Proposition4.4.2, we have L∞(F ) = 5.

Example 4.4.4. Let v = (v1, v2, v3) be a primitive vector in Z3>1, such that 1 < v1 < v2 < v3.

Let F = (F1, F2, F3, F4) : C3 → C4 be the polynomial map, whose component functions aregiven by

F1(x1, x2, x3) = x3v21 + x3v1

2 + xv21 x

2v12 + xv2

1 ,

F2(x1, x2, x3) = x5v1v32 + x2v2v3

1 x3v1v23 − xv1v3

2 x4v1v23 − xv2

1 xv23 ,

F3(x1, x2, x3) = x7v1v23 − x2v2v3

1 x5v1v32 − x7v1v3

2 + x3v2v31 x2v1v2

3 ,

F4(x1, x2, x3) = x1x8v1v23 − xv2

1 + x2v12 .

It is easy to observe that F is pre-weighted homogeneous with respect to v with dv(F1) =3v1v2, dv(F3) = 5v1v2v3, dv(F3) = 7v1v2v3 and dv(F4) = v1 + 8v1v2v3. Moreover, A(v) = 3and dv(F1) = mindv(F1), dv(F2), dv(F3), dv(F4). It is immediate to see that

x ∈ C3 : qv(F2)(x) = qv(F3)(x) = qv(F4)(x) = 0 6⊆ x ∈ C3 : x3 = 0,

since (0, 1, 1) ∈ x ∈ C3 : qv(F2)(x) = qv(F3)(x) = qv(F4)(x) = 0 \ x ∈ C3 : x3 = 0.Then by Proposition 4.4.2, we obtain that 3v1v2/v3 = L∞(qv(F )) and 3v1v2/v3 6 L∞(F ).Moreover, we claim that L∞(F ) = 3v1v2/v3. Indeed, the meromorphic curve γ : C\0 → C3


defined by γ(t) = (0, t−v2 , t−v3) is not the zero, with ord(γ) = −v3. A direct computationshows that F γ(t) = (t−3v1v2 , 0, 0, t−2v1v2) is not the zero curve. Let β > 3v1v2/v3. Then wehave

limt→0

‖F (γ(t))‖‖γ(t)‖β = lim

t→0

‖(t−3v1v2 , 0, 0, t−2v1v2)‖‖(0, t−v2 , t−v3)‖β = 0 (4.41)

where the last equality follows from β > 3v1v2/v3. In particular L∞(F ) 6 3v1v2/v3. ThereforeL∞(F ) = 3v1v2/v3.

4.5 Index of polynomial maps

In this section we show a result concerning the index of real polynomial vector fields.This result will follow as a consequence of the argument of the proof of Theorem 4.2.4. IfF : Rn → Rn is a real polynomial map such that F−1(0) is finite, then we denote by ind(F )the index of F , that is

ind(F ) =∑

x∈F−1(0)indx(F ),

where indx(F ) denotes the topological index of F at x.The next theorem is motivated by results obtained by following authors Cima-Gasull-

Mañosas [18], Gutitérrez-Ruas [36] and Bivià-Ausina [5].

Theorem 4.5.1. Let F,G : Rn → Rn be polynomial maps such that F and F + G have afinite number of zeros. Let us assume that

(i) F is strongly adapted to Γ+;

(ii) Ei(F, Γ+) 6= ∅, for all i = 1, . . . , n;

(iii) `(w,Gi) > `(w,Fi), for all w ∈ F0(Γ+), i = 1, . . . , n.

Thenind(F ) = ind(F +G).

Proof. Let us consider the homotopy H : [0, 1]×Rn → Rn defined by H(t, x) = F (x)+ tG(x).Let Ht = (Ht,1, . . . , Ht,n) : Rn → Rn be given by Ht(x) = H(t, x), for all x ∈ Rn and allt ∈ [0, 1]. We claim that there exists a uniform Łojasiewicz inequality at infinity for thefamily of maps Htt∈[0,1]. That is, there exist some constants M,α > 0 such that

‖x‖α 6 C‖Ht(x)‖, (4.42)


for all pair (t, x) ∈ [0, 1] × Rn such that ‖x‖ > M . As a consequence we would obtain thatH−1t (0) is contained in the open ball B(0;M), for all t ∈ [0, 1]. In particular H(t, x) 6= 0 for

all (t, x) ∈ [0, 1] × ∂B(0;M), where ∂B(0;M) denotes the boundary of B(0;M). This factimplies ind(F ) = ind(F +G), as a consequence of a known result about the invariance of theindex by homotopies Teorem 1.4.7.

From Γ+ we can construct a subdivision of Rn into simplicial cones as explained in theproof of Theorem 4.2.4. Let us keep the notation introduced in Subsection 4.2, before Lemma4.2.1. Let us fix a cone σ ∈ Σ(n). Let a1(σ), . . . , an(σ) be the primitive generators of σ. Letus consider the decomposition of a1(σ), . . . , an(σ) as in (4.8) and (4.9).

Analogous to (4.14) we can consider, for each i = 1, . . . , n and each t ∈ [0, 1], the polyno-mial H∗σ,t,i ∈ K[yσ,1, . . . , yσ,n] such that

Ht,i πσ(yσ) = y`(a1(σ),Ht,i)σ,1 · · · y`(ar(σ),Ht,i)

σ,r ·H∗σ,t,i(yσ),

for all yσ ∈ Wσ.By hypothesis we have `(w,Gi) > `(w,Fi), for all w ∈ F0(Γ+), i = 1, . . . , n. This implies

`(aj(σ), Ht,i) = `(aj(σ), Fi), for all i = 1, . . . , n, j = 1, . . . , r. Hence the principal parts of Fiand of Ht,i with respect to any subset of F0(Γ+) coincide, for all i = 1, . . . , n (see Definition4.1.1) and

Ei(Ht, Γ+) = Ei(F, Γ+) 6= ∅, (4.43)

for all i = 1, . . . , n and all t ∈ [0, 1] (the sets Ei(F, Γ+) are non-empty by hypothesis).Let us see that there exists some constant M > 0 such that

infyσ∈π−1

σ (VM )t∈[0,1]

supi|H∗σ,t,i(yσ)| 6= 0 (4.44)

for all σ ∈ Σ(n). If we assume the opposite then there exists a cone σ ∈ Σ(n) and a sequence(tm, ym)m>1 ⊆ [0, 1]×Wσ verifying that πσ(ym)m>1 →∞ and H∗σ,tm,i(ym)m>1 → 0, forall i = 1, . . . , n. Analogous to the proof of Proposition 4.2.2, let us consider a limit point(t,y) = (t,y1, . . . ,yn) ∈ [0, 1]×W σ of (tm, ym)m>1. By continuity we haveH∗σ,t,i(y) = 0, forall i = 1, . . . , n. Moreover, since πσ(ym)m>1 → ∞, we have that the set J0 = j : yj = 0is non-empty.

Following the same procedure as in the proof of Proposition 4.2.2 (see (4.18)) we obtain

pJ0(Ht,i) πσ(y) =∏j /∈J0

y`(aj(σ),Ht,i)

σ,j H∗σ,t,i(y) = 0, (4.45)


for all i = 1, . . . , n, where y = (y1, . . . , yn) is the point defined by

yj =

yj, if j /∈ J0,

1, if j ∈ J0.

Then we have a contradiction, since pJ0(Ht,i) = pJ0(Fi), for all i = 1, . . . , n, and F isstrongly adapted to Γ+. Hence relation (4.44) holds for some M > 0 and all σ ∈ Σ(n). As aconsequence, since Ei(Ht, Γ+) = Ei(F, Γ+) 6= ∅, for all i = 1, . . . , n and all t ∈ [0, 1], we canreproduce inequalities (4.21)-(4.25), by replacing Fi by Ht,i and taking k = ei, to obtain thatfor all i = 1, . . . , n, there exists a constant Ci > 0 such that

|xi|bi 6 Ci‖F (x) + tG(x)‖, (4.46)

for all x ∈ Rn such that ‖x‖ > M and all t ∈ [0, 1], where bi = sup Ei, for all i = 1, . . . , n(see (4.33)).

Let b0 = minb1, . . . , bn and C0 = maxC1, . . . , Cn. We can assume that M >√n.

Hence ‖x‖ >M implies maxi |xi| > 1. Taking maxi at both sides of (4.46), we obtain

‖x‖b0 6 (√n)b0 max |xi|b0 6 (

√n)b0C0‖F (x) + tG(x)‖

for all t ∈ [0, 1] and all x ∈ Rn such that ‖x‖ > M . Then there exists a uniform Łojasiewiczinequality at infinity for the family of maps Htt∈[0,1] and the result follows, as explained atthe beginning of the proof.

We observe from chapter 1, that if F : Cn → Cn is a polynomial map, then I(F ) denotesthe ideal of C[x1, . . . , xn] generated by the component functions of F and denote by FR themap R2n → R2n obtained from F under the identification x+ iy ↔ (x, y) between C and R2.We remark that the proof of Theorem 4.5.1 also works to deduce the following result.

Theorem 4.5.2. Let F,G : Cn → Cn be polynomial maps such that

(i) F is strongly adapted to Γ+;

(ii) Ei(F, Γ+) 6= ∅, for all i = 1, . . . , n;

(iii) `(w,Gi) > `(w,Fi), for all w ∈ F0(Γ+), i = 1, . . . , n.

Then F and F +G have a finite number of zeros and

dimCC[x1, . . . , xn]

I(F ) = dimCC[x1, . . . , xn]I(F +G) .


Proof. Conditions (i), (ii) and (iii) imply that F +G also satisfy the hypothesis of Corollary4.3.3. Then L∞(F ) and L∞(F + G) are positive numbers. This implies that F−1(0) and(F +G)−1(0) are compact and hence finite. The same proof of Theorem 4.5.1 works to obtainthat there is a uniform Łojasiewicz inequality for the homotopy H : [0, 1]×Cn → Cn definedby H(t, z) = F (z) + tG(z), for all (t, z) ∈ [0, 1]×Cn. That is, there exist some constants M ,α > 0 such that

‖z‖α 6 C‖F (z) + tG(z)‖,

for all z ∈ Cn such that ‖z‖ > M and all t ∈ [0, 1]. In particular, by Theorem 1.4.7 thismeans that ind(FR) = ind(FR +GR).

It is well known (see for instance Teorem 1.5.4) that

dimCC[x1, . . . , xn]

I(F ) =∑

z∈F−1(0)dimC

On,zIz(F ) .

where On,z denotes the germ of analytic function germs (Cn, z)→ C and Iz(F ) is the ideal ofOn,z generated by the germs of the component functions of F at z, for any z ∈ Cn. It is alsoknown by Theorem 1.4.9 that, if z = x + iy ∈ F−1(0), then dimCOn,z/Iz(F ) = ind(x,y)(FR)(see also, for instance[16, p. 146] or [30, p. 15]). Then the result follows.

If F : Rn → Rn denotes a real polynomial map, then we denote by FC : Cn → Cn themap obtained from F by complexifying the variables.

Remark 4.5.3. Under the hypothesis of Theorem 4.5.1, if we assume that FC is stronglyadapted to Γ+, then L∞(FC) > 0, by Corollary 4.3.3, and consequently the zero set of themaps F and F +G are finite.

Example 4.5.4. Let us consider the polynomial map F = (F1, F2, F3) : R3 → R3, where

F1(x1, x2, x3) = xa11 + xa1

1 xb12 + xa1

1 xc13 + αxa1

1 xb12 x

c13 ,

F2(x1, x2, x3) = xb22 + xa2

1 xb22 + xb2

2 xc23 + βxa2

1 xb22 x

c23 ,

F3(x1, x2, x3) = xc33 + xa3

1 xc33 + xb3

2 xc33 + γxa3

1 xb32 x

c33 .

where α, β, γ are mutually different non-zero real numbers and the supports of the abovepolynomials are contained in Z3

>1. Let g be the function defined by g(x1, x2, x3) = x1 + x2 +x3 + x1x2 + x1x3 + x2x3 + x1x2x3 and let Γ+ = Γ+(g) ⊆ R3. Then we observe that Γ+ is


convenient and F0(Γ+) = −e1,−e2,−e3. A direct computation shows that

`(−e1, F1) = −a1, `(−e2, F1) = −b1, `(−e3, F1) = −c1, `(e1, F1) = a1,

`(−e1, F2) = −a2, `(−e2, F2) = −b2, `(−e3, F2) = −c2, `(e2, F2) = b2,

`(−e1, F3) = −a3, `(−e2, F2) = −b3, `(−e3, F2) = −c3, `(e3, F3) = c3.

and `(ei, Fj) = 0, for all i, j ∈ 1, 2, 3, such that i 6= j.It is straightforward to check that FC is strongly adapted to Γ+ and Ei(F, Γ+) 6= ∅,

for i = 1, 2, 3. In particular 0 < minai, bi, ci : i = 1, 2, 3 6 L∞(F ), by Corollary 4.3.3.Therefore F−1(0) is finite. Let G = (G1, G2, G3) : R3 → R3 be a polynomial map such thatsupp(Gi) is contained in the cube (k1, k2, k3) ∈ Z3

>0 : k1 < ai, k2 < bi, k3 < ci, for i = 1, 2, 3.Then (F +G)−1(0) is also finite and ind(F ) = ind(F +G), by Theorem 4.5.1.

Example 4.5.5. Let a, b, c ∈ Z>1 and let us consider the map F = (F1, F2, F3) : R3 → R3

given byF (x1, x2, x3) = (xa1 + xb2 + xa1x

b2, x

a1x

b2 + xc3, x

c3).

Let Γ+ = Γ+(F ). Then it is immediate to see that FC is strongly adapted to Γ+ andEi(F, Γ+) 6= ∅, for all i = 1, 2, 3. Hence F−1(0) is finite. Then, by Theorem 4.5.1, anypolynomial map G : R3 → R3 such that supp(Gi) is contained in the interior of Γ+ verifiesthat (F +G)−1(0) is finite and ind(F ) = ind(F +G).

In the following example shows that the condition L∞(F ) = r0(Γ+(F )) does not implythat F is Newton non-degenerate at infinity.

Example 4.5.6. Let ai, bi, ci,∈ Z3>1, for all i = 1, 2 such that a1 < a2, b1 < b2 and c1 < c2.

Let F = (F1, F2, F3) : C3 → C3 be the polynomial map, whose component functions are givenby

F1(x1, x2, x3) = xa11 + xb1

2 + xc13 + α1x

a21 x

b22 + β1x

b22 x

c23 + γ1x

a21 x

c23 ,

F2(x1, x2, x3) = xa11 + xb1

2 + xc13 + α2x

a11 x

b22 + β2x

b12 x

c23 + γ2x

a21 x

c13 ,

F3(x1, x2, x3) = xa11 + xb1

2 + xc13 + xa1

1 xb12 x

c13 .

where α1, α2, β1, β2, γ1, γ2 are mutually different non-zero real numbers and the supports ofthe above polynomials are contained in Z3

>1.


The following picture shows the Newton polyhedron at infinity of F .

(a ,b ,0)

(a ,b ,0)

(0,b ,c )(0,b ,c )

(a ,0,c )

(a ,0,c )

a

c

b

1

1

12 1

2 2

1 2

2 221

2 2

0

x3

x1

x2

Let Γ+ = Γ+(g) be the Newton polyhedron in R3 given in Example 4.5.4. Then we observethat Γ+ is convenient and F(Γ+) = e1, e2, e3,−e1,−e2,−e3. A direct computation showsthat

`(−e1, F1) = −a2, `(−e2, F1) = −b2, `(−e3, F1) = −c2,

`(−e1, F2) = −a2, `(−e2, F2) = −b2, `(−e3, F2) = −c2,

`(−e1, F3) = −a1, `(−e2, F2) = −b1, `(−e3, F2) = −c1,

and `(ei, Fj) = 0, for all i, j = 1, 2, 3.It is straightforward to check that F is adapted to Γ+ and since Fi is convenient for all

i = 1, 2, 3, by Proposition 4.1.9 we have that F is strongly adapted to Γ+. Moreover, it iseasy to check that Ei(F, Γ+) 6= ∅, for i = 1, 2, 3. In particular by Corollary 4.3.4, we concludethat L∞(F ) = mina1, b1, c1 = r0(Γ+(F )) > 0.

However, F is Newton degenerate at infinity. Indeed, let us consider the 2-dimensionalface ∆ of Γ+(F ) containing the points (a2, b2, 0), (a2, 0, c2) and (0, b2, c2). Then an immediatecomputation shows that (F1)∆ = α1x

a21 x

b22 + β1x

b22 x

c23 + γ1x

a21 x

c23 , and (F2)∆ = (F3)∆ = 0.

From this we conclude that F is Newton degenerate at infinity.


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Index

Łojasiewicz exponentof h with respect to I, 12

Łojasiewicz exponent at infinity, 19

adaptedsystem to Γ+, 13

adapted map, 73algebraic multiplicity of I at a, 17

convenient Newton polyhedron, 28convenient polynomial map, 29

degree of h with respect to v, 35dimension of a face ∆, 26

face of S, 26facets of Γ+, 26filtrating map associated to Γ+, 33finite analytic mapping, 14finite polynomial map, 18

Index of a smooth mapping, 15integral closure of I, 6integrally closed, 6

local Newton polyhedron, 8

meromorphic curve, 20Milnor number, 32Minkowski sum of Γ1

+, . . . , Γp+ , 27

multiplicity of I with respect to M , 6multiplicity of R, 6Multiplicity of F, 18

Newton non-degenerate at infinity, 30Newton number, 32Newton polyhedron at infinity, 25Newton polyhedron at infinity of h, 28non-degenerate, 76non-degenerate with respect to Γ+, 35

polynomial map, 14pre-weighted homogeneous, 36primitive vector, 27principal part

of h at infinity with respect to v, 35of h over ∆, 35of h with respect to J , 72of h with respect to w, 72

quasi-special polynomials, 47

special polynomial, 46strongly adapted maps, 75support of h, 28

Topological degree of mapping, 15

vertices of Γ+, 26

weighted-homogeneous, 35

106

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Non-degeneracy of polynomial maps with respect to global ... · Orientador: Prof. Dr. Marcelo José Saia Coorientador: Prof. Dr. Carles Bivià-Ausina USP – São Carlos Agosto de