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

Abstract

Let F : K → K 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 K → K, of the notion of integral closure of ideals in the ring of analytic function germs (K, 0)→ K. 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 Γ+ ⊆ R. We obtain an approximation to S0(F ) when F is strongly adapted to Γ+, which is a condition expressed in terms of the faces of Γ+ and the principal parts at infinity of F . The local version of this problem has been studied by Bivia-Ausina [4] and Saia [71]. Our result about the estimation of S0(F ) allows us to give a lower estimate for the Łojasiewicz exponent at infinity of a given polynomial map with compact zero set. As a consequence of our study of Łojasiewicz exponents at infinity we have also obtained a result about the uniformity of the Łojasiewicz exponent in deformations of polynomial maps K → K. Consequently we derive a result about the invariance of the global index of real polynomial maps R → R. As particular cases of the condition of F being adapted to Γ+ 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 : K → K in terms of the set S0((F, 1)), where (F, 1) : K → 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 .