On model complete differential fields

Hrushovski, Ehud; Itai, Masanori

doi:10.1090/s0002-9947-03-03264-1

Cited by 27 publications

(44 citation statements)

References 23 publications

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“…The definability is claimed in [9] but done more clearly in [20,Section 2.4]. Of course, quantifier elimination for DCF 0 allows us to take the definition to be quantifier-free.…”

Section: 2mentioning

confidence: 99%

Turing Degree Spectra of Differentially Closed Fields

Marker¹,

Miller²

2017

J. symb. log.

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Abstract. The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that if this spectrum contained a low degree, then it would contain the degree 0. From these results we conclude that the spectra of differentially closed fields of characteristic 0 are exactly the jump-preimages of spectra of automorphically nontrivial graphs.

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“…The definability is claimed in [9] but done more clearly in [20,Section 2.4]. Of course, quantifier elimination for DCF 0 allows us to take the definition to be quantifier-free.…”

Section: 2mentioning

confidence: 99%

Turing Degree Spectra of Differentially Closed Fields

Marker¹,

Miller²

2017

J. symb. log.

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“…x ? ' , and 5F C % x g" 7 , and the two middle vertical arrows are restrictions of n x n" and p x p" to 3S.…”

Section: Corollary 32 (Acfa) If Some Very Dense L"-type P In {Stf mentioning

confidence: 99%

“…Given an L-definable si of Morley degree 1 and an Indefinable finite {and, therefore, dominant) rational f: s/ -> sf 7 , suppose that some very dense type in (stf, / ) " is grouplike. Then there are an irreducible group W and a finite dominant rational group homomorphism <f>: W -> W 7 , an irreducible W and a finite dominant rational a: <& -> <&", and L-definable finite dominant rational % and p such that the following diagram commutes: V If §5. Characterization for curves in ACFA.…”

Section: What Were We Chasing After?mentioning

confidence: 99%

Grouplike minimal sets in ACFA AND inT_A

Medvedev

2010

J. symb. log.

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This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in T A . The thesis concerns minimal formulae of the form x ∈ A ∧ σ(x) = f (x) for an algebraic curve A and a dominant rational function f : A → σ(A).These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f ) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to T A , the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A T A analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.

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“…In this example, there is only one 'problematic' nonorthogonality class, so that the resulting collapsed structure will still be of rank ω, but will not be multidimensional. Similar ideas were already used in the constructions of [21].…”

Section: The Geometric Structure Of T ∞mentioning

confidence: 89%

“…To the best of my knowledge the question of definability of orthogonality in the context of amalgamation constructions has been addressed explicitly only in [21] (in a different setting) where it is the main technical tool required for the construction, and in § 7 of [15]. In the original works of Hrushovski's non-orthogonality of simple pre-algebraic n-types amounted to the action of Sym(n) on the variables of p, as can be inferred, from the proof of the Algebraic Amalgamation Lemma of [18].…”

Section: Systems Of Standard Geometriesmentioning

confidence: 99%

Some questions concerning Hrushovski's amalgamation constructions

Hasson

2008

J. Inst. Math. Jussieu

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In order to construct a counterexample to Zilber's conjecture—that a strongly minimal set has a degenerate, affine or field-like geometry—Ehud Hrushovski invented an amalgamation technique which has yielded all the exotic geometries so far. We shall present a framework for this construction in the language of standard geometric stability and show how some of the recent constructions fit into this setting. We also ask some fundamental questions concerning this method.

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On model complete differential fields

Cited by 27 publications

References 23 publications

Turing Degree Spectra of Differentially Closed Fields

Turing Degree Spectra of Differentially Closed Fields

Grouplike minimal sets in ACFA AND inT_A

Some questions concerning Hrushovski's amalgamation constructions

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On model complete differential fields

Cited by 27 publications

References 23 publications

Turing Degree Spectra of Differentially Closed Fields

Turing Degree Spectra of Differentially Closed Fields

Grouplike minimal sets in ACFA AND inTA

Some questions concerning Hrushovski's amalgamation constructions

Contact Info

Product

Resources

About

Grouplike minimal sets in ACFA AND inT_A