Quantitative equidistribution for the solutions of systems of sparse polynomial equations

TL;DR

This paper demonstrates that solutions of certain sparse polynomial systems are approximately uniformly distributed near the unit polycircle, extending classical univariate root distribution results to multivariate cases with applications in root counting and distribution analysis.

Contribution

It generalizes the Erdős-Turán distribution result to multivariate Laurent polynomial systems with bounded coefficients, providing new bounds and distribution insights.

Findings

Solutions are approximately equidistributed near the unit polycircle.

Bounds on the number of real roots of Laurent polynomial systems.

Analysis of root distribution for random and integer coefficient systems.

Abstract

For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},..., x_n^{\pm1}] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic n-th dimensional complex torus of the system of equations f_1=\dots=f_n=0, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case a classical result due to Erdos and Turan on the distribution of the arguments of the roots of a univariate polynomial. We apply this result to bound the number of real roots of a system of Laurent polynomials, and to study the asymptotic distribution of the roots of systems of Laurent polynomials with integer coefficients, and of random systems of Laurent polynomials with complex coefficients.