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

TL;DR

This work extends the $\mathbb{A}^{1}$-Brouwer degree framework implemented in the Macaulay2 package A1BrouwerDegrees to finite étale algebras, enabling transfers in Grothendieck–Witt theory and broadening computations beyond the base fields previously supported. It also introduces unstable Grothendieck–Witt theory, including a concrete unstable GW group $\text{GW}^{u}(L)$ and practical constructors, isomorphism tests, and decompositions, thereby enabling explicit unstable class manipulation. The authors provide robust methods for computing global and local unstable $\mathbb{A}^{1}$-degrees via Bézoutian determinants and Newton matrices, and establish a Poincaré–Hopf-type relation at rational points that links local contributions to the global unstable degree. Collectively, these updates expand the scope and practicality of explicit $\mathbb{A}^{1}$-Brouwer degree calculations in motivic homotopy theory, expanding applicability to more general zero configurations and enabling precise algebraic invariants to be computed in Macaulay2 $\boldsymbol{2.0}$ with accompanying documentation.

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 étale algebras, allowing the computation of transfers along finite étale 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.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Proposition 3
• proof
• Remark 4