On canonical bases and internality criteria

Moosa, Rahim; Pillay, Anand

doi:10.1215/ijm/1254403721

Cited by 22 publications

(43 citation statements)

References 15 publications

(24 reference statements)

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“…This higher order analogue of Kolchin's differential tangent space was introduced by Pillay and Ziegler in [13] where it was used to prove what is now called the Canonical Base Property; a strong property which, among other things, gives a quick and Zariski-geometry-free proof of the Zilber dichotomy for differentially closed fields. Here we study differential jet spaces in their own right, and prove that they satisfy a certain strengthening of internality to the constants introduced implicitly by the third author and Pillay in [9], and then refined and formalised in [8]. This strengthening of internality is the differential analogue of a property that complexanalytic jet spaces enjoy, and went provisionally by the name "being Moishezon" in [8].…”

Section: Introductionmentioning

confidence: 88%

Differential-Algebraic Jet Spaces Preserve Internality to the Constants

Chatzidakis¹,

Harrison-Trainor²,

Moosa³

2015

J. symb. log.

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Abstract. Suppose p is the generic type of a differential-algebraic jet space to a finite dimensional differential-algebraic variety at a generic point. It is shown that p satisfies a certain strengthening of almost internality to the constants. This strengthening, which was originally called "being Moishezon to the constants" in [8] but is here renamed preserving internality to the constants, is a model-theoretic abstraction of the generic behaviour of jet spaces in complex-analytic geometry. An example is given showing that only a generic analogue holds in the differential-algebraic case: there is a finite dimensional differential-algebraic variety X with a subvariety Z that is internal to the constants, such that the restriction of the differential-algebraic tangent bundle of X to Z is not almost internal to the constants.

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Section: Introductionmentioning

confidence: 88%

Differential-Algebraic Jet Spaces Preserve Internality to the Constants

Chatzidakis¹,

Harrison-Trainor²,

Moosa³

2015

J. symb. log.

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“…The projective line in CCM internalises finite covers, while the field of constants in DCF 0 does not. For the former see Fact 3.1 of [26], and for the latter note that the generic type of δ(x 2 − b) = 0 where b is a fixed non-constant is almost internal but not internal to the field of constants (see [33,Lemma 3.1]).…”

Section: (3) Every Completion Of T a Is Simple Andmentioning

confidence: 99%

Model theory of compact complex manifolds with an automorphism

Bays

Hils

Moosa

2017

Trans. Amer. Math. Soc.

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Motivated by possible applications to meromorphic dynamics, and generalising known properties of difference-closed fields, this paper studies the theory CCMA of compact complex manifolds with a generic automorphism. It is shown that while CCMA does admit geometric elimination of imaginaries, it cannot eliminate imaginaries outright: a counterexample to 3-uniqueness in CCM is exhibited. Finite-dimensional types are investigated and it is shown, following the approach of Pillay and Ziegler, that the canonical base property holds in CCMA. As a consequence the Zilber dichotomy is deduced: finitedimensional minimal types are either one-based or almost internal to the fixed field. In addition, a general criterion for stable embeddedness in T A (when it exists) is established, and used to determine the full induced structure of CCMA on projective varieties, simple nonalgebraic complex tori, and simply connected nonalgebraic strongly minimal manifolds.

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“…Recently, relative versions of one-basedness have come into play. The main example is the canonical base property (CBP), which originates in [9] and [10], was formally defined in [4], and also studied by Chatzidakis [1], and in a more general framework by Palacín and Wagner [5], where (a weak version of) the CBP is indeed treated as a generalization of one-basedness.…”

Section: Introductionmentioning

confidence: 99%

On definable Galois groups and the strong canonical base property

Palacín

Pillay

2017

J. Math. Log.

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In [3], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that T has the canonical base property in a strong form; " internality to" being replaced by "algebraicity in". In the current paper we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than [3], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that 1-based groups are rigid.

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On canonical bases and internality criteria

Cited by 22 publications

References 15 publications

Differential-Algebraic Jet Spaces Preserve Internality to the Constants

Differential-Algebraic Jet Spaces Preserve Internality to the Constants

Model theory of compact complex manifolds with an automorphism

On definable Galois groups and the strong canonical base property

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