Intersection multiplicity, Milnor number and Bernstein's theorem

TL;DR

This paper characterizes when the Milnor number at the origin is minimal for polynomials with a given Newton diagram and extends Bernstein's theorem to solutions on unions of torus orbits, providing explicit conditions and solutions.

Contribution

It provides explicit characterizations of minimal Milnor numbers and maximum isolated zeros for polynomial systems supported on given polytopes, extending classical results.

Findings

Characterization of when the Milnor number is minimal for a polynomial.

Explicit conditions for systems with maximum isolated zeros supported on given polytopes.

Extension of Bernstein's theorem to solutions on unions of torus orbits.

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.