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arXiv · 2311.06044

Uniform existential definitions of valuations in function fields in one variable

Karim Johannes Becher·Nicolas Daans·Philip Dittmann
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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.

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Karim Johannes Becher, Nicolas Daans, Philip Dittmann. 2025-09-23. Uniform existential definitions of valuations in function fields in one variable. https://arxiv.org/abs/2311.06044

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