Universality of Persistence of Random Polynomials

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 $\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}((t-s)\mathrm{/}2{)}^{\alpha +1}$. 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.