  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 $\alpha$. 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 $\alpha=d/2-1$.
