Bézoutians and the 𝔸1-degree

Brazelton, Thomas; McKean, Stephen; Pauli, Sabrina

doi:10.2140/ant.2023.17.1985

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Bézoutians and the $\mathbb{A}^1$-degree

Thomas Brazelton, Stephen McKean and Sabrina Pauli

Vol. 17 (2023), No. 11, 1985–2012

DOI: 10.2140/ant.2023.17.1985

Abstract

We prove that both the local and global $𝔸^{1}$ -degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that the Bézoutian bilinear form, the Scheja–Storch form, and the $𝔸^{1}$ -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.

Keywords

Bézoutian, Brouwer degree, A1-degree, motivic homotopy

Mathematical Subject Classification

Primary: 14F42

Secondary: 55M25

Milestones

Received: 11 April 2022

Revised: 29 September 2022

Accepted: 20 January 2023

Published: 3 October 2023

Authors

Thomas Brazelton
Department of Mathematics University of Pennsylvania Philadelphia, PA United States

Stephen McKean
Department of Mathematics Harvard University Cambridge, MA United States

Sabrina Pauli
Fakultät für Mathematik Universität Duisburg-Essen Essen Germany

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Vol. 17, No. 11, 2023