Specialization of Difference Equations and High Frobenius Powers

TL;DR

This paper develops a valuation theory for valued fields with a contracting automorphism, establishing model-theoretic properties and linking to Frobenius actions, enabling new approaches in difference algebraic geometry.

Contribution

It introduces a transformal valuation theory, proves the existence of a model companion with quantifier elimination, and connects to Frobenius actions on valued fields.

Findings

Model companion is decidable and admits quantifier elimination.

Transformal wild ramification is controlled by torsors over vector groups.

The theory aligns with the limit of Frobenius actions on algebraically closed valued fields.

Abstract

We study valued fields equipped with an automorphism $\sigma$ which is locally infinitely contracting in the sense that $\alpha\ll\sigma\alpha$ for all $0<\alpha\in\Gamma$. We show that various notions of valuation theory, such as Henselian and strictly Henselian hulls, admit meaningful transformal analogues. We prove canonical amalgamation results, and exhibit the way that transformal wild ramification is controlled by torsors over generalized vector groups. Model theoretically, we determine the model companion: it is decidable, admits a simple axiomatization, and enjoys elimination of quantifiers up to algebraically bounded quantifiers. The model companion is shown to agree with the limit theory of the Frobenius action on an algebraically closed and nontrivially valued field. This opens the way to a motivic intersection theory for difference varieties that was previously available only in characteristic zero. As a first consequence, the class of algebraically closed valued fields equipped with a distinguished Frobenius $x\mapsto x^{q}$ is decidable, uniformly in $q$.