Where are the zeroes of a random p-adic polynomial?

Abstract

We study the repartition of the roots of a random p-adic polynomial in an algebraic closure of Qp.We prove that the mean number of roots generating a fixed finite extension K of Qp depends mostly on the discriminant of K, an extension containing less roots when it gets more ramified. We prove further that, for any positive integer r, a random p-adic polynomial of sufficiently large degree has about r roots on average in extensions of degree at most r.Beyond the mean, we also study higher moments and correlations between the number of roots in two given subsets of Qp (or, more generally, of a finite extension of Qp). In this perspective, we notably establish results highlighting that the roots tend to repel each other and quantify this phenomenon.