The \(\mathbb{A}^1\)-degree is an analogue in algebraic geometry of the Brouwer degree from topology. It was constructed by F. Morel [in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035–1059 (2006; Zbl 1097.14014)] using motivic homotopy theory and assigns a bilinear form to an endomorphism of affine \(n\)-space. The \(\mathbb{A}^1\)-degree has many applications in the fast-growing field of refined enumerative geometry, which aims to obtain refinements of results in enumerative geometry in terms of quadratic forms. One can then often deduce known results by taking e.g. ranks or signatures.
In this paper, the authors give a general algebraic formula for computing \(\mathbb{A}^1\)-degrees in practice. Let \(k\) be a field which is not of characteristic \(2\) and let \(f:\mathbb{A}^n_k\to\mathbb{A}^n_k\) be a morphism. They show that the local \(\mathbb{A}^1\)-degree of \(f\) at a closed point where \(f\) has an isolated zero can be computed as a bilinear form determined by a Bézoutian form, the latter being relatively easy to compute explicitly. If all the zero’s of \(f\) are isolated, they show that the global \(\mathbb{A}^1\)-degree of \(f\) can be computed in terms of a Bézoutian form as well. The statement is deduced from work of Bachmann-Wickelgren [T. Bachmann and K. Wickelgren, J. Inst. Math. Jussieu 22, No. 2, 681–746 (2023; Zbl 1515.14037)] and some computations.
This gives an explicit way to compute \(\mathbb{A}^1\)-degrees, which was later used heavily by Brazelton and others [N. Borisov et al., “\(\mathbb{A}^1\)-Brouwer degrees in Macaulay2”, Preprint, arXiv:2312.00106] in their implementation of computations of \(\mathbb{A}^1\)-degrees in Macaulay2. In the paper itself, there are also several applications: The authors compute an \(\mathbb{A}^1\)-degree which wouldn’t have been easily accessible otherwise (Example 7.2), they prove a formula for the quadratic Euler characteristic of a Grassmanian which was folklore before (section 8) and they use their results to describe various calculation rules for \(\mathbb{A}^1\)-degrees (section 6).
The paper is very well-written and provides a good introduction to Bézoutians, Scheja-Storch forms and \(\mathbb{A}^1\)-degrees. At the end, there is a nice picture of a modified Pascal’s triangle which is used to illustrate a part of the computation for the quadratic Euler characteristic of a Grassmanian.