Duke Mathematical Journal

The class of Eisenbud–Khimshiashvili–Levine is the local A1-Brouwer degree

Jesse Leo Kass and Kirsten Wickelgren

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Given a polynomial function with an isolated zero at the origin, we prove that the local A1-Brouwer degree equals the Eisenbud–Khimshiashvili–Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes, together with associated arithmetic information, by enriching Milnor’s equality between the local degree of the gradient and the number of nodes into which a hypersurface singularity bifurcates to an equality in the Grothendieck–Witt group.

Article information

Duke Math. J., Volume 168, Number 3 (2019), 429-469.

Received: 9 October 2017
Revised: 17 October 2018
First available in Project Euclid: 16 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 55M25: Degree, winding number

A1 degree A1 enumerative geometry Eisenbud–Levine/Khimshiashvili signature formula Milnor number


Kass, Jesse Leo; Wickelgren, Kirsten. The class of Eisenbud–Khimshiashvili–Levine is the local $\mathbf{A}^{1}$ -Brouwer degree. Duke Math. J. 168 (2019), no. 3, 429--469. doi:10.1215/00127094-2018-0046. https://projecteuclid.org/euclid.dmj/1547607998

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