Factorization statistics and the twisted Grothendieck-Lefschetz formula

TL;DR

This paper explores a new connection between polynomial factorization over finite fields and the cohomology of configuration spaces in three dimensions, extending previous work from two dimensions.

Contribution

It introduces a novel link between factorization statistics and the cohomology of configuration spaces in D, generalizing earlier results in D.

Findings

Establishes a relationship between factorization statistics and cohomology in D.

Extends the framework of Church, Ellenberg, and Farb to higher dimensions.

Provides new insights into the structure of polynomials over finite fields.

Abstract

We announce recent results on a connection between factorization statistics of polynomials over a finite field and the structure of the cohomology of configurations in $\mathbb{R}^3$ as a representation of the symmetric group. This connection parallels a result of Church, Ellenberg, and Farb relating factorization statistics of squarefree polynomials and the cohomology of configurations in $\mathbb{R}^2$.