Uniform existential definitions of valuations in function fields in one variable

2023/11/10 by Karim Johannès Becher, Becher, Karim Johannes, Nicolas Daans +3 · 1 citation
Computer Science · Mathematics · #11E81 #12F20 #12L05 #12L99 #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2311.06044

openalex publication_date 2023/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study function fields of curves over a base field K which is either a global field or a large field having a separable field extension of degree divisible by 4. We show that, for any such function field, Hilbert's 10th Problem has a negative answer, the valuation rings containing K are uniformly existentially definable, and finitely generated integrally closed K-subalgebras are definable by a universal-existential formula. In order to obtain these results, we develop further the usage of local-global principles for quadratic forms in function fields to definability of certain subrings. We include a first systematic presentation of this general method, without restriction on the characteristic.

Cited by

• First-order definability of Campana Points and Darmon Points in algebraic function fields in one variable over number fields

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