Zero distribution of random sparse polynomials

TL;DR

This paper investigates the asymptotic distribution of zeros of random Laurent polynomials with support in dilated polytopes, providing a localized Bernstein-Kouchnirenko type result for various coefficient distributions.

Contribution

It introduces a quantitative localized version of the Bernstein-Kouchnirenko Theorem for random Laurent polynomials with broad classes of coefficient distributions.

Findings

Asymptotic zero distribution characterized for large degree polynomials.

Established a localized Bernstein-Kouchnirenko type theorem.

Applicable to diverse probability distributions with bounded density.

Abstract

We study asymptotic zero distribution of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope $P$ as their degree grow. We consider a large class of probability distributions including the ones induced from i.i.d. random coefficients whose distribution law has bounded density with logarithmically decaying tails as well as moderate measures defined over the projectivized space of Laurent polynomials. We obtain a quantitative localized version of Bernstein-Kouchnirenko Theorem.