Topological fields with a generic derivation

Kovacsics, Pablo Cubides; Point, Françoise

doi:10.1016/j.apal.2022.103211

Cited by 2 publications

(3 citation statements)

References 43 publications

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“…In [15], we initiated a study of definable groups in closed ordered differential fields (see [20]), and more generally in differential expansions of o-minimal structures, p-adically closed fields, pseudo-finite fields of characteristic 0, or topological fields which are models of an open theory (as in [13]).…”

Section: The Setting and Main Resultsmentioning

confidence: 99%

“…Theorem 4. 13 Let T be the theory of pseudo-finite fields, and let Γ be a finitedimensional definable group in K ⊧ T ∂ . Then there exists a K-algebraic D-group (H, s) such that Γ is virtually definably isogenous over K to the K-points of (H, s) ∂ .…”

Section: The Case Of Pseudo-finite Fieldsmentioning

confidence: 99%

“…We now return to the proof of Theorem 4. 13. By [15,Theorem 5.11], there exist an L-definable group G and a definable group embedding ∶ σ ∶ Γ → G. Furthermore, every generic L-type of G is realized by an element in σ(Γ) and, in addition, there is a covering of G by finitely many L(K)definable sets X i , i = 1, .…”

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confidence: 99%

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On definable groups andD-groups in certain fields with a generic derivation

Peterzil,

Pillay,

Point

2024

Can. J. Math.-J. Can. Math.

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We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$ , the model companion of an o-minimal ${\mathcal {L}}$ -theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007). We generalize Buium’s notion of an algebraic D-group to ${\mathcal {L}}$ -definable D-groups, namely $(G,s)$ , where G is an ${\mathcal {L}}$ -definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$ -definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form $$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$ for some ${\mathcal {L}}$ -definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$ ). We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic $0$ .

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Section: The Setting and Main Resultsmentioning

confidence: 99%

Section: The Case Of Pseudo-finite Fieldsmentioning

confidence: 99%

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On definable groups andD-groups in certain fields with a generic derivation

Peterzil,

Pillay,

Point

2024

Can. J. Math.-J. Can. Math.

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Generic Derivations on Algebraically Bounded Structures

FORNASIERO,

TERZO

2024

J. symb. log.

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Let ${\mathbb K}$ be an algebraically bounded structure, and let T be its theory. If T is model complete, then the theory of ${\mathbb K}$ endowed with a derivation, denoted by $T^{\delta }$ , has a model completion. Additionally, we prove that if the theory T is stable/NIP then the model completion of $T^{\delta }$ is also stable/NIP. Similar results hold for the theory with several derivations, either commuting or non-commuting.

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Topological fields with a generic derivation

Cited by 2 publications

References 43 publications

On definable groups andD-groups in certain fields with a generic derivation

On definable groups andD-groups in certain fields with a generic derivation

Generic Derivations on Algebraically Bounded Structures

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