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

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 $1/p$.

arXiv comment22 pages; to appear in the Proceedings of the London Mathematical Society; this version includes further details in the proof of Lemma 3.5