Analytic and pseudo-analytic structures (a survey)

Source: arXiv

ABSTRACT The paper is an extended version of the talk in the Logic Colloquium-2000 at Paris. We discuss a series of results and problems around Hrushovski's construction of counter-examples to the Trichotomy conjecture.

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    ABSTRACT: Let ℳ=(M,<) be a linearly ordered structure. ℳ is o-minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitrary o-minimal ℳ can induce on a neighborhood of any a in M. Roughly said, either (i) a is trivial (technical term), or (ii) a has a convex neighborhood on which ℳ induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which ℳ induces the structure of an expansion of a real closed field. The proof uses “geometric calculus” which allows to recover a differentiable structure by purely geometric methods.
    Proceedings of the London Mathematical Society 01/1998; 77(3):481-523. · 1.15 Impact Factor
  • Logic Colloquium '85, Orsay, France; 01/1985
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    ABSTRACT: A dierence eld is a eld with a distinguished automorphism . This paper studies the model theory of existentially closed dierence elds. We introduce a dimension theory on formulas, and in particular on dierence equa- tions. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional for- mulas when the characteristic is 0.
    Transactions of the American Mathematical Society 01/1999; 351(8). · 1.02 Impact Factor

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