Quantifier elimination for algebraic $D$-groups

Abstract. We introduce the categories of algebraic σ-varieties and σ-groups over a difference field (K, σ). Under a "linearly σ-closed" assumption on (K, σ) we prove an isotriviality theorem for σ-groups. This theorem immediately yields the key lemma in a proof of the Manin-Mumford conjecture. The present paper crucially uses ideas of Pilay and Ziegler (2003) but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski's work (2001) and are also closely related to papers of Roessler (2002 and2004).

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“…The proof proceeds as in the proof of the analogous Theorem 3.2 of [7]. The relevant lemmas were proved in Section 2.…”

Section: Piotr Kowalski and Anand Pillay Finite Field And A Homomorpmentioning

confidence: 95%

“…In [7], the authors studied algebraic D-varieties and D-groups and proved a "Chevalley-type theorem": the image of a D-constructible subset of an algebraic D-group under a D-homomorphism is also D-constructible. Here D-constructible means Boolean combination of D-closed.…”

Section: Piotr Kowalski and Anand Pillay Finite Field And A Homomorpmentioning

confidence: 99%

On algebraic 𝜎-groups

Kowalski

Pillay

2006

Trans. Amer. Math. Soc.

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“…Let A be a nonempty definable set of (U, G) Remark 4.4. 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]).…”

Section: A Quantifier Elimination Results For Parameterized D-groupsmentioning

confidence: 82%

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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“…So q is the Kolchin generic type of ( V , s) ♯ (K) over k alg , for some irreducible component V of V . Isotriviality of (V, s) implies isotriviality of ( V , s), and this means that q is internal to the constants K δ , see for example [16,Fact 2.6]. By stability, this implies that the binding group G = Aut(q/k alg (K δ )) is type-definable over k alg , see for instance [11,Appendix B].…”

Section: Maximum D-subvarietiesmentioning

confidence: 99%

D-groups and the Dixmier–Moeglin equivalence

2018

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A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for D-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if R is a commutative affine Hopf algebra over a field of characteristic zero, and A is an Ore extension to which the Hopf algebra structure extends, then A satisfies the classical Dixmier-Moeglin equivalence. Along the way it is shown that all such A are Hopf Ore extensions in the sense of [Brown et al., "Connected Hopf algebras and iterated Ore extensions", Journal of Pure and Applied Algebra, 219 (6), 2015].

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Quantifier elimination for algebraic $D$-groups

Cited by 20 publications

References 13 publications

On algebraic 𝜎-groups

On algebraic 𝜎-groups

On parameterized differential Galois extensions

D-groups and the Dixmier–Moeglin equivalence

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