Existentially defining valuations in function fields over large fields

2025/12/04 by Nicolas Daans, Daans, Nicolas
Mathematics · #11R52 #11R58 #12L05 (Secondary) #12L99 (Primary) #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2512.04896

openalex publication_date 2025/12/04 · openalex created_date 2025/12/06 · openalex updated_date 2026/07/28

Abstract

Let K be a large field such that K[√(-1)] is not algebraically closed and F/K a function field in one variable. Extending techniques and results from earlier work with Becher and Dittmann, we show that every valuation ring on F containing K is existentially definable in the language of rings with parameters from F. As a consequence, using a known reduction technique, we obtain the undecidability of the existential theory of F in the language of rings with appropriately chosen parameters.

Citations

• Decidability of polynomial equations over function fields in positive characteristic
• Existential rank and essential dimension of diophantine sets
• Universally and existentially definable subsets of global fields
• Hilbert's Tenth Problem for rational function fields over p-adic fields
• Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields

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