The \(\mathbb{A}^1\)-Brouwer degree is an analogue of the classical Brouwer degree from differential topology in the world of motivic homotopy theory. Over the last years, it had numerous applications to the fast-growing field of refined enumerative geometry, which aims to find “quadratic refinements” of classical results in enumerative geometry. Such refinements are quadratic forms, from which one can recover classical results e.g. by taking signatures or ranks, and possibly find new results over other fields.
In the paper by T. Brazelton et al. [Homology Homotopy Appl. 23, No. 1, 243–255 (2020; Zbl 1456.14027)], a method is developed for computing the \(\mathbb{A}^1\)-degree of a map \(f:\mathbb{A}^n_k\to\mathbb{A}^n_k\) from affine \(n\)-space over a field \(k\) to itself, at a closed point \(p\) with separable and finite residue extension \(k(p)/k\). Namely, one has \[ \deg^{\mathbb{A}^1}_p(f) = \text{Tr}_{k(p)/k}(\deg^{\mathbb{A}^1}_{\tilde{p}}(f_{k(p)})) \] where \(\text{Tr}_{k(p)/k}\) is the trace map on quadratic forms induced by the field extension \(k(p)/k\) and \(\tilde{p}\) is the canonical \(k(p)\)-rational point over \(p\).
In the present paper, McKean and Brazelton extend this result to the situation where \(k(p)/k\) can be a general finite extension. Two problems will generally arise in this situation: the rank of a resulting form is too big and the trace form is degenerate. One example in which such things happen is when \(k = \mathbb{F}_p(t)\) for \(p\neq 2\) a prime number, and one considers the map \(f:\mathbb{A}^1_k\to\mathbb{A}^1_k, x\mapsto x^p-t\), and tries to compute the \(\mathbb{A}^1\)-degree at the closed point defined by the ideal \((x^p-t)\subset k[x]\).
The authors use two concepts of transfer to overcome these issues: the geometric transfer \(\tau_k^{k(p)}(t)\) (which is the pushforward of a Scharlau form if \(k/k(p)\) is a finite separable extension) and the cohomological transfer \(\text{Tr}_k^{k(p)}\) (which is the composition with the field trace for a finite separable extension). Correspondingly, there are two lifts of the map \(f\): a geometric one \(f_g\), and a cohomological one \(f_c\) (which agrees with the base change \(f_{k(p)}\) in the separable case so that one can recover the original result). Their main theorem states that for a morphism \(f:\mathbb{A}^1_k\to\mathbb{A}^1_k\) with isolated root at a closed point \(p\), one has that \[ \deg^{\mathbb{A}^1}_p(f) = \tau_k^{k(p)}(t)\deg^{\mathbb{A}^1}_{\tilde{p}}(f_g) = \text{Tr}_k^{k(p)}\deg^{\mathbb{A}^1}_{\tilde{p}}(f_c). \] As a corollary, one can bound the non-hyperbolic part of the \(\mathbb{A}^1\)-degree of a polynomial map. Furthermore, the authors exhibit all scaled trace and Scharlau forms as \(\mathbb{A}^1\)-degrees.
The article gives a good exposition of the methods used, coming from motivic homotopy theory and matrix tools (in particular, Hankel forms and Horner bases). A lot of intuition and motivation is given for the main result and the definitions. The proofs also provide an application of McKeans generalized Bézoutians. At the end of the paper, there is also an appendix explaining the diagonalization arguments pictorially, which is adapted to readers having the paper in black-white or in color.