Lifts, transfers, and degrees of univariate maps

TL;DR

This paper develops methods to compute local -degrees of univariate maps at points with inseparable residue fields, extending previous techniques and applying the framework to trace forms of number fields.

Contribution

It introduces a new approach for calculating local -degrees at inseparable points using lifts of polynomials and transfers, expanding the computational toolkit.

Findings

Computed local -degrees at inseparable points via polynomial lifts.

Connected trace forms of number fields to local -degrees.

Discussed the six functor formalism in the context of -degrees.

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.