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

Abstract

Abstract We study the distribution of the roots of a random p-adic polynomial in an algebraic closure of ${\mathbb Q}_p$ . We prove that the mean number of roots generating a fixed finite extension K of ${\mathbb Q}_p$ depends mostly on the discriminant of K, an extension containing fewer roots when it becomes 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 ${\mathbb Q}_p$ (or, more generally, of a finite extension of ${\mathbb Q}_p$ ). In this perspective, we notably establish results highlighting that the roots tend to repel each other and quantify this phenomenon.