The density of polynomials of degree $n$ over $\mathbb{Z}_p$ having exactly $r$ roots in $\mathbb{Q}_p$

TL;DR

This paper calculates the probability that a random polynomial over rac{}p_p has exactly r roots in rac{}p_p, revealing a rational function invariant under p to 1/p substitution.

Contribution

It provides an explicit formula for the probability distribution of roots of polynomials over rac{}p_p, highlighting a symmetry property.

Findings

Probability expressed as a rational function of p

Invariant under p to 1/p transformation

Explicit formula for root count distribution

Abstract

We determine the probability that a random polynomial of degree $n$ over $\mathbb{Z}_p$ has exactly $r$ roots in $\mathbb{Q}_p$, and show that it is given by a rational function of $p$ that is invariant under replacing $p$ by $1/p$.