Homology, Homotopy and Applications

Volume 23 (2021)

Number 1

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

Pages: 243 – 255

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a13


Thomas Brazelton (Department of Mathematics, David Rittenhouse Laboratory,University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Robert Burklund (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Stephen McKean (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Michael Montoro (Department of Mathematics, The University at Buffalo, Buffalo, New York, U.S.A.)

Morgan Opie (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


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 relating the Scheja–Storch form and the local $\mathbb{A}^1$-degree.


motivic, homotopy, enumerative, geometry, trace, degree

2010 Mathematics Subject Classification

14F42, 55M25, 55P42

Copyright © 2020, Thomas Brazelton, Robert Burklund, Stephen McKean, Michael Montoro and Morgan Opie. Permission to copy for private use granted.

Received 27 January 2020

Received revised 29 June 2020

Accepted 30 June 2020

Published 14 October 2020