Decidability of positive characteristic tame Hahn fields in L_t

2021/08/09 by Victor Lisinski, Lisinski, Victor
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Cellular Automata and Applications #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2108.04132

openalex publication_date 2021/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that any positive characteristic tame Hahn field \mathbbF((t^Γ)) containing t is decidable in L_t, the language of valued fields with a constant symbol for t, if \mathbbF and Γ are decidable. In particular, we obtain decidability of \mathbbF_p((t^1/p^∞)) and \mathbbF_p((t^ℚ)) in L_t. This uses a new AKE-principle for equal characteristic tame fields in L_t, building on work by Kuhlmann, together with Kedlaya's work on finite automata and algebraic extensions of function fields. In the process, we obtain an AKE-principle for tame fields in mixed characteristic.

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