Lifts, transfers, and degrees of univariate maps

Abstract

One can compute the local ${\mathbb{A}}^1$-degree at points with separable residue field by base changing, working rationally, and post-composing with the field trace. We show that for endomorphisms of the affine line, one can compute the local ${\mathbb{A}}^1$-degree at points with inseparable residue field by taking a suitable lift of the polynomial and transferring its local degree. We also discuss the general set-up and strategy in terms of the six functor formalism. As an application, we show that trace forms of number fields are local ${\mathbb{A}}^1$-degrees.

arXiv comment29 pages, 2 figures. Comments welcome!