Universality of Persistence of Random Polynomials

TL;DR

This paper studies the probability that certain random polynomials have no real zeros, revealing a universal asymptotic behavior that extends previous results to broader coefficient distributions.

Contribution

It generalizes existing results by removing moment and distributional restrictions, establishing a universal asymptotic probability for no real zeros in random polynomials.

Findings

Probability asymptotically behaves as n^{-2(b_α + b_0)}

Generalizes previous results to non-Gaussian, finite-moment coefficients

Confirms a conjecture for the case α=0

Abstract

We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d. sequence of mean-zero, variance-1 random variables, multiplied by an $\frac{\alpha}{2}$-regularly varying sequence for $\alpha>-1$. We show that the probability of no real zeros is asymptotically $n^{-2(b_{\alpha}+b_0)}$, where $b_{\alpha}$ is the persistence exponents of a mean-zero, one-dimensional stationary Gaussian processes with covariance function as $\mathrm{sech}((t-s)/2)^{\alpha+1}$. Our work generalizes the previous results of Dembo et al. [DPSZ02] and Dembo \& Mukherjee [DM15] by removing the requirement of finite moments of all order or Gaussianity. In particular, in the special case $\alpha = 0$, our findings confirm a conjecture by Poonen and Stoll [PS99, Section 9.1] concerning random polynomials with i.i.d. coefficients.