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BĂ©zoutians and the $\mathbb{A}^1$-degree

Thomas Brazelton, Stephen McKean and Sabrina Pauli

Vol. 17 (2023), No. 11, 1985–2012

We prove that both the local and global 𝔸1-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 𝔸1-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.

BĂ©zoutian, Brouwer degree, A1-degree, motivic homotopy
Mathematical Subject Classification
Primary: 14F42
Secondary: 55M25
Received: 11 April 2022
Revised: 29 September 2022
Accepted: 20 January 2023
Published: 3 October 2023
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

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