We study finite-dimensional groups definable in models of the theory $\mathit{RCF}_{\partial }$ of real closed fields with a generic derivation (also known as CODF, the theory of closed ordered differential fields [28]). We prove that any such group $\Gamma $ definably embeds in a “canonical” semialgebraic group $G$.
We explain how our methods work in the general context of strongly model complete theories $T$ of large “geometric” fields with a generic derivation, which includes the cases where $T$ is the theory of pseudofinite fields and $T = \mathrm{Th}({\mathbb{Q}}_{p})$. We also give a general theorem on recovering a definable group from generic data in the context of geometric theories.
Finally we extend the methods to $o$-minimal theories with a generic derivation, due to Fornasiero and Kaplan, [10], and open theories of topological fields with a generic derivation, due to Cubides-Kovacsics and the third author, [7].
Nous étudions les groupes définissables, de dimension finie dans les modèles de la théorie $\mathit{RCF}_{\partial }$ des corps réels-clos munis d’une dérivée générique (aussi connue sous l’acronyme CODF, la théorie des corps ordonnés différentiellement clos [28]). Nous montrons qu’un tel groupe $\Gamma $ se plonge dans un groupe semi-algébrique $G$ « canonique ».
Nous expliquons comment nos méthodes se généralisent aux théories fortement modèles-complètes de corps « géométriques », larges, munis d’une dérivée générique. Ceci comprend les cas de la théorie des corps pseudo-finis et de la théorie du corps des nombres $p$-adiques. Nous montrons également, dans le contexte des théories géométriques, comment récupérer un groupe définissable à partir de données génériques.
Enfin nous étendons nos méthodes aux théories o-minimales avec une dérivée générique, traitées par Fornasiero et Kaplan, [10], et aux théories « ouvertes » de corps topologiques avec une dérivée générique, étudiées par Cubides-Kovacsics et le troisième auteur, [7].
Ya’acov Peterzil 1 ; Anand Pillay 2 ; Françoise Point 3
CC-BY-ND 4.0
Ya’acov Peterzil; Anand Pillay; Françoise Point. On definable groups in real closed fields with a generic derivation, and related structures. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2026), pp. 21-52. doi: 10.5802/pmb.68
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author = {Ya{\textquoteright}acov Peterzil and Anand Pillay and Fran\c{c}oise Point},
title = {On definable groups in real closed fields with a generic derivation, and related structures},
journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
pages = {21--52},
year = {2026},
publisher = {Presses universitaires de Franche-Comt\'e},
doi = {10.5802/pmb.68},
language = {en},
url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.68/}
}
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