Jet spaces of varieties over differential and difference fields

Pillay, Anand; Ziegler, Martin

doi:10.1007/s00029-003-0339-1

Cited by 58 publications

(117 citation statements)

References 20 publications

(23 reference statements)

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“…The CBP has other interesting consequences, which can be found in [77], in [98], in [65] and in [16]. It is still open for minimal types in SCF e , and more generally in most fields of positive characteristic (maybe with additional structure) which are not algebraically closed.…”

Section: 7mentioning

confidence: 99%

Model Theory of Fields With Operators – a Survey

Chatzidakis¹

2015

Logic Without Borders

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Section: 7mentioning

confidence: 99%

Model Theory of Fields With Operators – a Survey

Chatzidakis¹

2015

Logic Without Borders

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“…However, because we need a more direct relationship to our underlying field K we take a different, more analytic, approach. The method we are using below is based on the techniques of Camapana and Fujiki in the setting of compact complex manifolds, as interpreted in [18] by Pillay and Ziegler and, in the o-minimal setting, in [14].…”

Section: The Structure A(m ) and Mild Manifoldsmentioning

confidence: 99%

Mild manifolds and a non-standard Riemann existence theorem

Peterzil

Starchenko

2008

Sel. math., New ser.

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Abstract. Let R be an o-minimal expansion of a real closed field R and K be the algebraic closure of R. In earlier papers we investigated the notions of R-definable K-holomorphic maps, K-analytic manifolds and their K-analytic subsets. We call such a K-manifold mild if it eliminates quantifiers after endowing it with all K-analytic subsets. Examples are compact complex manifolds and non-singular algebraic curves over K.We examine here basic properties of mild manifolds and prove that when a mild manifold M is strongly minimal and not locally modular then M is bi-holomorphic with a non-singular algebraic curve over K.

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“…As in [PZ,2.4], for n big enough the Gauss map as defined in Fact 2.8 is an embedding modulo Stab(X). Therefore, the rational D-map X → Gr r (J n e (G)) factors through a rational embedding X/Stab(X) → Gr(J n e (G)).…”

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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Jet spaces of varieties over differential and difference fields

Cited by 58 publications

References 20 publications

Model Theory of Fields With Operators – a Survey

Model Theory of Fields With Operators – a Survey

Mild manifolds and a non-standard Riemann existence theorem

Quantifier elimination for algebraic $D$-groups

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