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

TL;DR

This paper investigates the distribution and correlation of roots of random p-adic polynomials, revealing how roots are influenced by extension discriminants and exhibit repulsion, with implications for understanding p-adic root behavior.

Contribution

It provides new insights into the distribution, moments, and correlations of roots of random p-adic polynomials, including root repulsion phenomena and dependence on extension discriminants.

Findings

Mean number of roots depends on the discriminant of the extension.

Roots tend to repel each other, showing negative correlation.

Large degree polynomials have about r roots in extensions of degree at most r.

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.