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arXiv · 2506.19411

Counting rational points on transcendental curves in valued fields

Floris Vermeulen
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Abstract

We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to Cluckers--Comte--Loeser from $p$-adic fields to arbitrary valued fields of mixed characteristic. Our methods rely on parametrizations, where we avoid the usage of $r$-th power maps, combined with the determinant method.

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Floris Vermeulen. 2025-06-24. Counting rational points on transcendental curves in valued fields. https://arxiv.org/abs/2506.19411

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