Kirsten Wickelgren: Motivic local degree, A1-Milnor numbers, and an arithmetic count of the lines on a cubic surface

ABSTRACT: A celebrated result of Eisenbud--Kimshaishvili--Levine computes the local degree of a smooth function \(f : R^n \to R^n\) with an isolated zero at the origin. Given a polynomial function with an isolated zero at the origin, we prove that the local A1-Brouwer degree equals the degree quadratic form of Eisenbud--Khimshiashvili--Levine, answering a question posed by David Eisenbud in 1978. This talk will present this result and then discuss applications to the study of singularities and an arithmetic count of the lines on a cubic surface. This is joint work with Jesse Leo Kass.