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Grouplike minimal sets in ACFA AND in TA

Published online by Cambridge University Press:  12 March 2014

Alice Medvedev
Alice Medvedev*
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street Chicago, IL 60607-7045, USA. E-mail: [email protected]
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Abstract

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 TA. The thesis concerns minimal formulae of the form x Є 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 TA, 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 TA 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 4 , December 2010 , pp. 1462 - 1488
Copyright
Copyright © Association for Symbolic Logic 2010

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References

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