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

Thomas Brazelton, Stephen McKean, Sabrina Pauli

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.

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

Abstract

We prove that both the local and global -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 -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.

Paper Structure

This paper contains 15 sections, 22 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Background
  4. Notation and conventions
  5. Bézoutians
  6. The Scheja--Storch bilinear form
  7. Local decomposition
  8. Proof of Theorem \ref{['thm:main-thm']}
  9. Computing the Bézoutian bilinear form
  10. Calculation rules
  11. Examples
  12. Application: the $\mathbb{A}^1$-Euler characteristic of Grassmannians
  13. The $\mathbb{A}^1$-Euler characteristic
  14. The $\mathbb{A}^1$-Euler characteristic of Grassmannians
  15. Modified Pascal's triangle for $\chi^{\mathbb{A}^1}(\operatorname{Gr}_k(r,n))$

Key Result

Theorem 1.2

Let $\operatorname{char}{k}\neq 2$. Let $f: \mathbb{A}^n_k \to \mathbb{A}^n_k$ have an isolated zero at a closed point $\mathfrak{m}$. Then $\beta_{f,\mathfrak{m}}$ is isomorphic to the local $\mathbb{A}^1$-degree of $f$ at $\mathfrak{m}$. If we further assume that all the zeros of $f$ are isolated,

Figures (3)

  • Figure 1: More examples of $\chi^{\mathbb{A}^1}\left(\operatorname{Gr}_k(r,n)\right)$
  • Figure 2: Addition rules for modified Pascal's triangle
  • Figure 3: Modified Pascal's triangle for $\chi^{\mathbb{A}^1}(\operatorname{Gr}_k(r,n))$

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 3.1
  • Example 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 51 more