Intersection multiplicity, Milnor number and Bernstein's theorem

Abstract

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which completes works of Kushnirenko (Invent. Math., 1976) and Wall (J. Reine Angew. Math., 1999). Given a fixed collection of n convex integral polytopes in R^n, we also give an explicit characterization of systems of n polynomials supported at these polytopes which have the maximum number (counted with multiplicity) of isolated zeroes on k^n, or more generally, on a union of torus orbits on k^n; this completes the program (undertaken by many authors including Khovanskii (Funkcional. Anal. i Prilozen, 1978), Huber and Sturmfels (Discrete Comput. Geom., 1997), Rojas (J. Pure Appl. Algebra, 1999)) of the extension to k^n of Bernstein's theorem (Funkcional. Anal. i Prilozen, 1975) on number of solutions of n polynomials on (k^*)^n. Our solutions to these two problems are connected by the computation of the intersection multiplicity at the origin of n hypersurfaces determined by n generic polynomials.