Transfers and Unstable Degrees in the $\mathbb{A}^{1}$-Brouwer Degrees Package for Macaulay2

TL;DR

This paper presents an update to the Macaulay2 package A1BrouwerDegrees, enabling computations of transfers along finite étale extensions and enhancing the calculation of unstable -Brouwer degrees and Grothendieck--Witt classes.

Contribution

It extends the package to include transfer computations for finite étale algebras and introduces new features for unstable -Brouwer degree calculations and Grothendieck--Witt group manipulations.

Findings

Implemented transfer computations for finite étale extensions.

Enhanced calculation capabilities for unstable -Brouwer degrees.

Added features for manipulating classes in the unstable Grothendieck--Witt group.

Abstract

We describe a significant update to the Macaulay2 package A1BrouwerDegrees. We extend several methods in the previous version of the package to the setting of finite \'{e}tale algebras, allowing the computation of transfers along finite \'{e}tale extensions. Additionally, we implement a number of new features for the computation of unstable $\mathbb{A}^{1}$-Brouwer degrees and manipulation of classes in the unstable Grothendieck--Witt group.