Fox-Neuwirth cells, quantum shuffle algebras, and character sums of the resultant

TL;DR

This paper establishes an upper bound on character sums of the resultant over pairs of monic square-free polynomials using topological methods involving braid groups, homology, and quantum shuffle algebras.

Contribution

It introduces a novel topological approach to bounding character sums by computing braid group homology and relating it to quantum shuffle algebra bimodules.

Findings

Upper bound on character sums of the resultant

Homology computations for braid groups on multi-punctured planes

Vanishing range for homology of mixed braid groups with character-based local coefficients

Abstract

We give an upper bound on character sums of the resultant over pairs of monic square-free polynomials of given degrees, answering a question of Ellenberg and Shusterman in the quadratic case. Our approach is topological: we compute the homology of braid groups on multi-punctured planes and prove a vanishing range for the homology of mixed braid groups with rank-1 local coefficients associated to characters of finite fields. Our method involves constructing a cellular stratification for configuration spaces of multi-punctured planes and relating their twisted homology with more general exponential coefficients to the cohomology of certain bimodules over quantum shuffle algebras.