On the multiplicity of isolated roots of sparse polynomial systems

TL;DR

This paper derives formulas for the multiplicity of isolated roots in generic sparse polynomial systems, linking algebraic multiplicity to combinatorial invariants like mixed volumes and integrals.

Contribution

It introduces explicit formulas for root multiplicities based on the supports' combinatorial structure, extending classical results to affine isolated zeros.

Findings

Formulas for multiplicity of affine isolated zeros

Connection between multiplicity and mixed volumes

Application to general sparse polynomial systems

Abstract

We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein's theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions.