Random complex fewnomials, I

TL;DR

This paper studies the asymptotic distribution of zeros of random fewnomials with fixed monomial count and Gaussian coefficients, revealing a connection to Monge-Ampere measures and toric Kähler potentials.

Contribution

It introduces the concept of random fewnomials and derives limiting formulas for zero distributions as the degree grows large, linking to complex geometry.

Findings

Expected zero distribution converges to Monge-Ampere measure.

Limit measure described via a discrete Legendre transform.

Results apply to systems with equal number of equations and variables.

Abstract

We introduce several notions of `random fewnomials', i.e. random polynomials with a fixed number f of monomials of degree N. The f exponents are chosen at random and then the coefficients are chosen to be Gaussian random, mainly from the SU(m + 1) ensemble. The results give limiting formulas as N goes to infinity for the expected distribution of complex zeros of a system of k random fewnomials in m variables. When k = m, for SU(m + 1) polynomials, the limit is the Monge-Ampere measure of a toric Kaehler potential on CP^m obtained by averaging a `discrete Legendre transform' of the Fubini-Study symplectic potential at f points of the unit simplex in R^m.