During the period I was supported by the ERC grant, I wrote the paper [BMP] with Thomas Brazelton and Stephen McKean. We provide an algebraic formula for the \(\mathbb{A}^1\)-degree of a map \(f:\mathbb{A}^n\to \mathbb{A}^n\) with only isolated zeros in terms of the multivariate Bezoutian, which allows us to calculate the \(\mathbb{A}^1\)-degree without knowing the zeros of \(f\). This is a generalization of Cazanave's result in the case \(n=1\).

The multivariate Bezoutian can also be used to calculate the local \(\mathbb{A}^1\)-degee at an isolated zero yielding a formula for the local \(\mathbb{A}^1\)-degree without any restrictions on the residue field (it does not need to be separable over the base).

We also impleted code in Sage that computes the \(\mathbb{A}^1\)-degree and find some useful calculation rules for the \(\mathbb{A}^1\)-degree.

Project related publications

Project related preprints

[BMP] Thomas Brazelton, Stephen McKean, Sabrina Pauli, Bézoutians and the \(\mathbb{A}^1\)-degree. Preprint 2021. Publically available at https://services.math.duke.edu/~mckean/bezoutian.pdf

Works in progress


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