A Note on the Axioms for Differentially Closed Fields of Characteristic Zero

Pierce, David A.; Pillay, Anand

doi:10.1006/jabr.1997.7359

Cited by 45 publications

(55 citation statements)

References 3 publications

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“…Then a determines the right invariant vector field r a : G(U) → T (G)(U). As U is differentially closed, the main result of [7] gives…”

Section: Algebraic D-groupsmentioning

confidence: 99%

Algebraic D-groups and differential Galois theory

Pillay¹

2004

Pacific J. Math.

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We discuss various relationships between the algebraic Dgroups of Buium, 1992, and differential Galois theory. In the first part we give another exposition of our general differential Galois theory, which is somewhat more explicit than Pillay, 1998, and where generalized logarithmic derivatives on algebraic groups play a central role. In the second part we prove some results with a "constrained Galois cohomological flavor". For example, if G and H are connected algebraic D-groups over an algebraically closed differential field F , and G and H are isomorphic over some differential field extension of F , then they are isomorphic over some Picard-Vessiot extension of F . Suitable generalizations to isomorphisms of algebraic D-varieties are also given.

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“…Then a determines the right invariant vector field r a : G(U) → T (G)(U). As U is differentially closed, the main result of [7] gives…”

Section: Algebraic D-groupsmentioning

confidence: 99%

Algebraic D-groups and differential Galois theory

Pillay¹

2004

Pacific J. Math.

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“…In particular, K is algebraically closed, as is its field of constants. One use of differential-closedness is the following property, which is an instance of the "geometric axiom" for differentially closed fields (statement (ii) of Section 2 of [41]). …”

Section: 2mentioning

confidence: 99%

Poisson algebras via model theory and differential-algebraic geometry

Bell¹,

Launois²,

Sánchez³

et al. 2017

J. Eur. Math. Soc.

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Abstract. Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

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“…When D = {D} and ∆ = ∅, the parameterized prolongation τ D/∆ V is nothing more than the usual prolongation τ V used in ordinary differential algebraic geometry [11]. In this case, whenever V is defined over the constants we recover the tangent bundle of V .…”

Section: Preliminaries On Parameterized D-varieties and D-groupsmentioning

confidence: 99%

On parameterized differential Galois extensions

Sánchez

Nagloo

2016

Journal of Pure and Applied Algebra

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Abstract. We prove some existence results on parameterized strongly normal extensions for logarithmic equations. We generalize a result in [Wibmer, Existence of ∂-parameterized Picard-Vessiot extensions over fields with algebraically closed constants, J. Algebra, 361, 2012]. We also consider an extension of the results in [Kamensky and Pillay, Interpretations and differential Galois extensions, Preprint 2014] from the ODE case to the parameterized PDE case. More precisely, we show that if D and ∆ are two distinguished sets of derivations and (K D , ∆) is existentially closed in (K, ∆), where K is a D ∪ ∆-field of characteristic zero, then every (parameterized) logarithmic equation over K has a parameterized strongly normal extension.

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A Note on the Axioms for Differentially Closed Fields of Characteristic Zero

Cited by 45 publications

References 3 publications

Algebraic D-groups and differential Galois theory

Algebraic D-groups and differential Galois theory

Poisson algebras via model theory and differential-algebraic geometry

On parameterized differential Galois extensions

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