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

Contact: Pieter Moree (moree@mpim-bonn.mpg.de)

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.