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

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 Bézoutian. In particular, we show that the Bézoutian 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.

arXiv commentFixed a small typo

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• Motivic configurations on the line
• Compactly supported ${\mathbb{A}}^1$-Euler characteristics of symmetric powers of cellular varieties
• ${\mathbb{A}}^1$-Brouwer degrees in Macaulay2
• Trace maps in motivic homotopy and local terms