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

Abstract

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