p-adic root separation and the discriminant of integer polynomials

In this paper we investigate the following related problems: (A) the separation of $p$-adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with discriminant divisible by a (large) power of a fixed prime. One of the consequences of our findings is the existence, for all large $Q>1$, of $Q^{2\mathrm{/}n}$ integer irreducible polynomials $P$ of degree $n$ and height $\asymp Q$ with an almost prime power discriminant of maximal size, that is $\mathrm{\mid }D(P)\mathrm{\mid }\asymp Q^{2n-2}$ and $D(P)=p^k{C}_P$ with $C_P\in \mathbb{Z}$ satisfying $\mathrm{\mid }{C}_P\mathrm{\mid }\ll 1$. The method we use generalises the techniques used in the study of the real case [Beresnevich, Bernik and Götze, 2010 and 2016] and relies on a quantitative non-divergence estimate developed by Kleinbock and Tomanov.