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

• Autor(a) principal: Huarcaya, Jorge Alberto Coripaco
• Data de Publicação: 2015
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-04122015-094201/
• topic: Condições de não-degeneração; Global injectivity of polynomial maps; Index of real polynomials maps; Índice de aplicações polinomiais reais; Injectividade global de aplicações polinomiais; Lojasiewicz exponent at infinity; Multiplicidade de aplicações polinomiais; Multiplicity of polynomial maps; Newton polyhedra; Non-degeneracy conditions; Poliedros de Newton

Let F : Kn &rarr; Kp be a polynomial map, where K = R or C. Motivated by the characterization of the integral closure of ideals in the ring On by means of analytic inequalities proven by Lejeune-Teissier [46], we define the set Sp(F) of special polynomials with respect to F. The set Sp(F) can be considered as a counterpart, in the context of polynomial maps Kn &rarr; Kp, of the notion of integral closure of ideals in the ring of analytic function germs (~&lceil;+. In this work, we are mainly interested in the determination of the convex region S0(F) formed by the exponents of the special monomials with respect to F. Let us fix a convenient Newton polyhedron &lceil; + ~&sube; Rn. We obtain an approximation to S0</sub (F) when F is strongly adapted to ~&sube; +, which is a condition expressed in terms of the faces of ~&lceil;+ and the principal parts at infinity of F. The local version of this problem has been studied by Bivià-Ausina [4] and Saia [71]. Our result about the estimation of S0(F) allows us to give a lower estimate for the Lojasiewicz exponent at infinity of a given polynomial map with compact zero set. As a consequence of our study of ojasiewicz exponents at infinity we have also obtained a result about the uniformity of the ojasiewicz exponent in deformations of polynomial maps Kn &rarr; Kp. Consequently we derive a result about the invariance of the global index of real polynomial maps Rn &rarr; Rn. As particular cases of the condition of F being adapted to ~&lceil;+ there appears the class of Newton 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-degeneracy condition introduced by Kouchnirenko for polynomial functions. We characterize the Newton non-degeneracy at infinity condition of a given polynomial map F : Kn &rarr; Kp in terms of the set S0((F, 1)), where (F, 1) : Kn &rarr; Kp+1 is the polynomial map whose last component function equals 1. Motivated by analogous problems in local algebra we also derive some results concerning the multiplicity of F.