Differential Algebraic Groups of Finite Dimension

Buium, Alexandru

doi:10.1007/bfb0087235

Cited by 58 publications

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“…First, however, let us describe the target of s; compare [3], [2], [9]. Let (K, D) be a differential field, and V a variety over K (possibly reducible; it may be assumed K-irreducible.)…”

Section: Differential and Algebraic Varietiesmentioning

confidence: 99%

On model complete differential fields

Hrushovski

Itai

2003

Trans. Amer. Math. Soc.

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Abstract. We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE X. We show that this sub-family is usually definable (in particular if X lives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.

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“…First, however, let us describe the target of s; compare [3], [2], [9]. Let (K, D) be a differential field, and V a variety over K (possibly reducible; it may be assumed K-irreducible.)…”

Section: Differential and Algebraic Varietiesmentioning

confidence: 99%

On model complete differential fields

Hrushovski

Itai

2003

Trans. Amer. Math. Soc.

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“…It should be noted that in the ODE case algebraic D-groups (not necessarily defined over the constants) have quantifier elimination [5]. This result has an immediate extension to the PDE case when the parametric set of derivations ∆ is empty (these are the algebraic D-groups studied by Buium [1]). However, in general, ∆ = ∅, the situation is quite different.…”

Section: A Quantifier Elimination Results For Parameterized D-groupsmentioning

confidence: 80%

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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“…But the next fact will yield an even better result. It was proved by Buium for centerless G [B,Chapter 5,3.6].…”

Section: (Ii) X/stab(x) Is a D-subvariety Of G/stab(x)mentioning

confidence: 99%

Quantifier elimination for algebraic $D$-groups

Kowalski¹,

Pillay

2005

Trans. Amer. Math. Soc.

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Abstract. We prove that if G is an algebraic D-group (in the sense of Buium over a differentially closed field (K, ∂) of characteristic 0, then the first order structure consisting of G together with the algebraic D-subvarieties of G, G × G, . . . , has quantifier-elimination. In other words, the projection on G n of a D-constructible subset of G n+1 is D-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.

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Differential Algebraic Groups of Finite Dimension

Cited by 58 publications

References 0 publications

On model complete differential fields

On model complete differential fields

On parameterized differential Galois extensions

Quantifier elimination for algebraic $D$-groups

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