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

Manjul Bhargava, John Cremona, Tom Fisher, Stevan Gajović

Published 2021-01-23Version 1

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 $1/p$.