No zero-crossings for random polynomials and the heat equation

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 $(1,\mathrm{\infty })$ 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 $b_{\alpha }\in (0,\mathrm{\infty })$ 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 ${\varphi }_d(\mathbf{x},t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise ${\varphi }_d(\mathbf{x},0)$, we confirm that the probability of ${\varphi }_d(\mathbf{x},t)\mathrm{\ne }0$ for all $t\in [1,T]$, is $T^{-b_{\alpha }+o(1)}$, for $\alpha =d\mathrm{/}2-1$.