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arXiv · 2112.04592

Lifts, transfers, and degrees of univariate maps

Thomas Brazelton·Stephen McKean
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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.

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Thomas Brazelton, Stephen McKean. 2021-12-08. Lifts, transfers, and degrees of univariate maps. https://doi.org/10.7146/math.scand.a-134457

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