Semi-galois Categories II: An arithmetic analogue of Christol's theorem

TL;DR

This paper extends Christol's theorem to an arithmetic setting, characterizing algebraic power series over number fields using automata theory and Witt vectors, building on semi-galois categories.

Contribution

It introduces an arithmetic analogue of Christol's theorem, replacing polynomial rings with rings of integers and formal power series with Witt vectors.

Findings

Established an automata-theoretic characterization of algebraic power series over number fields.

Connected semi-galois categories with arithmetic analogues of classical theorems.

Explored related problems in the context of Witt vectors and number fields.

Abstract

In connection with our previous work on semi-galois categories, this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series over finite field is algebraic over the polynomial ring. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F_1) variant of Christol's theorem in the sense that it replaces the polynomial ring with the ring of integers of a number field and the ring of formal power series with the ring of Witt vectors. We also study some related problems.