No zero-crossings for random polynomials and the heat equation

Consider random polynomial .ni=0aixi of independent mean-zero normal coefficients ai, whose variance is a regularly varying function (in i) of order ..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.+o(1), and no roots in (1,.) with probability n.b0+o(1), hence for n even, it has no real roots with probability n.2b..2b0+o(1). Here, b.=0 when ...1 and otherwise b..(0,.) 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 .d(x,t) to the d-dimensional heat equation initiated by a Gaussian white noise .d(x,0), we confirm that the probability of .d(x,t).0 for all t.[1,T], is T.b.+o(1), for .=d/2.1.