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

Stevan Gajović (University of Groningen)

E1 05 (Leibniz-Saal)

Abstract

Let $f$ be a random polynomial in $Z p [x]$ of degree $n$ . We determine the density of such polynomials $f$ that have exactly $r$ roots in $Q p$ . We also determine the expected number of roots of monic polynomials $f$ in $Z p [x]$ 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.

Branching from number theory: p-adics in the sciences Branching from number theory: p-adics in the sciences

MPI für Mathematik in den Naturwissenschaften Leipzig E1 05 (Leibniz-Saal)

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Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Enis Kaya

University of Groningen

Avinash Kulkarni

Dartmouth College

Antonio Lerario

SISSA

Mima Stanojkovski

RWTH Aachen and Max Planck Institute for Mathematics in the Sciences