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 +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…