B\'ezoutians and the $\mathbb{A}^1$-degree

TL;DR

This paper establishes a connection between the $ ext{A}^1$-degree of affine space endomorphisms and multivariate Bézoutians, providing algebraic formulas for degrees in motivic homotopy theory.

Contribution

It proves that the $ ext{A}^1$-degree can be computed via Bézoutian bilinear forms, generalizing previous theorems to multivariate and local cases.

Findings

Bézoutian forms are isomorphic to the $ ext{A}^1$-degree for complete intersections.

The global theorem generalizes Cazanave's univariate case.

The local theorem extends Kass--Wickelgren's results on EKL forms.

Abstract

We prove that both the local and global $\mathbb{A}^1$-degree of an endomorphism of affine space can be computed in terms of the multivariate B\'ezoutian. In particular, we show that the B\'ezoutian bilinear form, the Scheja--Storch form, and the $\mathbb{A}^1$-degree for complete intersections are isomorphic. Our global theorem generalizes Cazanave's theorem in the univariate case, and our local theorem generalizes Kass--Wickelgren's theorem on EKL forms and the local degree. This result provides an algebraic formula for local and global degrees in motivic homotopy theory.