The trace of the local $\mathbf{A}^1$-degree

TL;DR

This paper establishes that the local ^1-degree of a polynomial at an isolated zero can be computed as a trace over the residue field, generalizing previous results and connecting motivic transfer concepts.

Contribution

It proves the trace formula for the local ^1-degree at isolated zeros with finite separable residue fields, extending prior work by Kass, Wickelgren, and Morel.

Findings

The local ^1-degree equals the trace over the residue field.

Generalization of Kass and Wickelgren's relation between Scheja-Storch form and ^1-degree.

Connection of motivic transfers with local degrees.

Abstract

We prove that the local $\mathbb{A}^1$-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local $\mathbb{A}^1$-degree over the residue field. This fact was originally suggested by Morel's work on motivic transfers and by Kass and Wickelgren's work on the Scheja-Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren's relating the Scheja-Storch form and the local $\mathbb{A}^1$-degree.