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.

Fixed a small typo

References in corpus (4)

• An arithmetic enrichment of Bézout's Theorem
• Quadratic types and the dynamic Euler number of lines on a quintic threefold
• Applications to A1-enumerative geometry of the A1-degree
• Computing A1-Euler numbers with Macaulay2

Cited by in corpus (4)

• 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