No zero-crossings for random polynomials and the heat equation
Consider random polynomial $\sum_{i=0}^na_ix^i$ of independent mean-zero normal coefficients $a_i$, whose variance is a regularly varying function (in $i$) of order $α$. 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 $n^{-b_α+o(1)}$, and no roots in $(1,\infty)$ with probability $n^{-b_0+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{-2b_α-2b_0+o(1)}$. Here, $b_α=0$ when $α\le-1$ and otherwise $b_α\in(0,\infty)$ 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 $ϕ_d({\mathbf{x}},t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise $ϕ_d({\mathbf{x}},0)$, we confirm that the probability of $ϕ_d({\mathbf{x}},t)\neq0$ for all $t\in[1,T]$, is $T^{-b_α+o(1)}$, for $α=d/2-1$.