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arXiv · 2103.16614

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

Thomas Brazelton·Stephen McKean·Sabrina Pauli
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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.

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Thomas Brazelton, Stephen McKean, Sabrina Pauli. 2021-03-30. B\'ezoutians and the $\mathbb{A}^1$-degree. https://doi.org/10.2140/ant.2023.17.1985

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