How many zeros of a random sparse polynomial are real?

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 Bürgisser, Ergür and Tonelli-Cueto showed that the expected number of real zeros of f in such cases is bounded by O( √ k log k). In this work, we improve the bound to O( √ 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 Ω( √ 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.