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

  • Stevan Gajović (University of Groningen, Groningen, Netherlands)
E1 05 (Leibniz-Saal)


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.


Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Enis Kaya

University of Groningen

Avinash Kulkarni

Dartmouth College

Antonio Lerario


Mima Stanojkovski

RWTH Aachen and Max Planck Institute for Mathematics in the Sciences