The density of polynomials of degree n over Zp that have exactly r roots in Qp

Let $f$ be a random polynomial in $Zp[x]$ of degree $n$. We determine the density of such polynomials $f$ that have exactly $r$ roots in $Qp$. We also determine the expected number of roots of monic polynomials $f$ in $Zp[x]$ of degree $n$, and more generally, the expected number of sets of exactly $d$ elements consisting of roots of such $f$. We show that these densities are rational functions in $p$, and discuss the remarkable symmetry phenomenon that occurs.

This is joint work Manjul Bhargava, John Cremona, and Tom Fisher.