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Corpus ID: 209140435

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

@article{Brazelton2019TheTO,
title={The trace of the local \$\mathbf\{A\}^1\$-degree},
author={T. Brazelton and Robert Burklund and S. McKean and Michael Montoro and Morgan Opie},
journal={arXiv: Algebraic Topology},
year={2019}
}

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.