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

TL;DR

This paper proves that for polynomial functions with isolated zeros, the local A1-Brouwer degree equals the Eisenbud-Khimshiashvili-Levine class, answering a question from 1978 and linking algebraic topology with algebraic geometry.

Contribution

It establishes the equality between the local A1-Brouwer degree and the Eisenbud-Khimshiashvili-Levine class for isolated zeros, resolving a long-standing question.

Findings

Proves the equality of local A1-Brouwer degree and Eisenbud-Khimshiashvili-Levine class.

Provides an application to counting nodes with arithmetic information.

Extends Milnor's equality to the Grothendieck-Witt group.

Abstract

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 degenerates to an equality in the Grothendieck-Witt group.