Counting polynomials over finite fields with given root multiplicities

TL;DR

This paper derives formulas for counting polynomials over finite fields with specified root multiplicities, revealing simple cases and connecting to configuration space topology, proposing new homological stabilization conjectures.

Contribution

It provides explicit formulas for counting polynomials with given root multiplicities and establishes a link to configuration spaces, suggesting new homological stabilization conjectures.

Findings

Formulas for counting polynomials with specified root multiplicities.

Identification of cases where formulas simplify to powers of q.

Proposed homological stabilization conjectures for configuration spaces.

Abstract

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an analogous result on configuration spaces in the Grothendieck ring of varieties, suggesting new homological stabilization conjectures for configuration spaces of the plane.