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Recursive spectra of strongly minimal theories satisfying the Zilber Trichotomy


Authors: Uri Andrews and Alice Medvedev
Journal: Trans. Amer. Math. Soc. 366 (2014), 2393-2417
MSC (2010): Primary 03C57, 03C10
DOI: https://doi.org/10.1090/S0002-9947-2014-05897-2
Published electronically: January 28, 2014
MathSciNet review: 3165643
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Abstract: We conjecture that for a strongly minimal theory $ T$ in a finite signature satisfying the Zilber Trichotomy, there are only three possibilities for the recursive spectrum of $ T$: all countable models of $ T$ are recursively presentable; none of them are recursively presentable; or only the zero-dimensional model of $ T$ is recursively presentable. We prove this conjecture for disintegrated (formerly, trivial) theories and for modular groups. The conjecture also holds via known results for fields. The conjecture remains open for finite covers of groups and fields.


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Additional Information

Uri Andrews
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Alice Medvedev
Affiliation: Department of Mathematics, The City College of New York, New York, New York 10031

DOI: https://doi.org/10.1090/S0002-9947-2014-05897-2
Received by editor(s): January 11, 2012
Received by editor(s) in revised form: June 6, 2012
Published electronically: January 28, 2014
Additional Notes: The second author was partially supported by NSF FRG DMS-0854998 grant.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.