No zero-crossings for random polynomials and the heat equation

Abstract

Consider random polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ of independent mean-zero normal coefficients $a_{i}$, whose variance is a regularly varying function (in $i$) of order $\alpha$. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in $[0,1]$ with probability $n^{-b_{\alpha}+o(1)}$, and no roots in $(1,\infty)$ with probability $n^{-b_{0}+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{-2b_{\alpha}-2b_{0}+o(1)}$. Here, $b_{\alpha}=0$ when $\alpha\le-1$ and otherwise $b_{\alpha}\in(0,\infty)$ is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution $\phi_{d}({\mathbf{x}},t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise $\phi_{d}({\mathbf{x}},0)$, we confirm that the probability of $\phi_{d}({\mathbf{x}},t)\neq0$ for all $t\in[1,T]$, is $T^{-b_{\alpha}+o(1)}$, for $\alpha=d/2-1$.