How many zeros of a random sparse polynomial are real?

TL;DR

This paper improves the upper bound on the expected number of real zeros of a sparse polynomial with Gaussian coefficients from O(√k log k) to O(√k), and shows this bound is tight, using an innovative approach based on the Edelman-Kostlan formulation.

Contribution

It provides a tighter bound on the expected real zeros of sparse Gaussian polynomials and introduces a new technique for analyzing zeros via the Edelman-Kostlan formulation.

Findings

Expected number of zeros bounded by O(√k)

Bound is tight with a matching lower bound

Technique recovers known bounds for dense polynomials

Abstract

We investigate the number of real zeros of a univariate $k$-sparse polynomial $f$ over the reals, when the coefficients of $f$ come from independent standard normal distributions. Recently B\"urgisser, Erg\"ur and Tonelli-Cueto showed that the expected number of real zeros of $f$ in such cases is bounded by $O(\sqrt{k} \log k)$. In this work, we improve the bound to $O(\sqrt{k})$ and also show that this bound is tight by constructing a family of sparse support whose expected number of real zeros is lower bounded by $\Omega(\sqrt{k})$. Our main technique is an alternative formulation of the Kac integral by Edelman-Kostlan which allows us to bound the expected number of zeros of $f$ in terms of the expected number of zeros of polynomials of lower sparsity. Using our technique, we also recover the $O(\log n)$ bound on the expected number of real zeros of a dense polynomial of degree $n$ with coefficients coming from independent standard normal distributions.