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.
Nous démontrons des bornes supérieures pour le nombre de points rationnels sur des courbes transcendantes dans des corps 1-h-minimaux arbitraires, analogues au théorème de dénombrement de Pila–Wilkie dans le cadre o-minimal. Ces résultats étendent ceux de Cluckers–Comte–Loeser des corps $p$-adiques à des corps valués arbitraires de caractéristique mixte. Nos méthodes reposent sur des paramétrisations, où nous évitons l’utilisation d’applications puissance $r$-ième, combinées à la méthode du déterminant.
Floris Vermeulen 1
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Floris Vermeulen. Counting rational points on transcendental curves in valued fields. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2026), pp. 61-69. doi: 10.5802/pmb.70
@article{PMB_2026____61_0,
author = {Floris Vermeulen},
title = {Counting rational points on transcendental curves in valued fields},
journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
pages = {61--69},
year = {2026},
publisher = {Presses universitaires de Franche-Comt\'e},
doi = {10.5802/pmb.70},
language = {en},
url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.70/}
}
TY - JOUR AU - Floris Vermeulen TI - Counting rational points on transcendental curves in valued fields JO - Publications mathématiques de Besançon. Algèbre et théorie des nombres PY - 2026 SP - 61 EP - 69 PB - Presses universitaires de Franche-Comté UR - https://pmb.centre-mersenne.org/articles/10.5802/pmb.70/ DO - 10.5802/pmb.70 LA - en ID - PMB_2026____61_0 ER -
%0 Journal Article %A Floris Vermeulen %T Counting rational points on transcendental curves in valued fields %J Publications mathématiques de Besançon. Algèbre et théorie des nombres %D 2026 %P 61-69 %I Presses universitaires de Franche-Comté %U https://pmb.centre-mersenne.org/articles/10.5802/pmb.70/ %R 10.5802/pmb.70 %G en %F PMB_2026____61_0
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