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Definable one dimensional structures in o-minimal theories

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Abstract

This is the first of two papers where we prove the Zil’ber trichotomy principle for one-dimensional structures definable in o-minimal ones.

Here we prove: Let N be a definable structure in an o-minimal structure M, with dim M (N) = 1. If N is stable then it is necessarily 1-based. Along the way, we develop a fine intersection theory for definable curves in o-minimal structures.

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Author notes

  1. Assaf Hasson

    Present address: Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva, 84120, Israel

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Assaf Hasson

  2. Departemento de Matemáticas, Universidad de los Andes, Cra. 1 No 18A-10, Bogotá, Colombia

    Alf Onshuus

  3. Department of Mathematics, University of Haifa, Haifa, 31905, Israel

    Ya’acov Peterzil

Authors
  1. Assaf Hasson
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  2. Alf Onshuus
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  3. Ya’acov Peterzil
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Corresponding author

Correspondence to Assaf Hasson.

Additional information

Supported by the EPSRC grant no. EP C52800X 1.

Partially supported by the EC FP6 through the Marie Curie Research Training Network MODNET (MRTN-CT-2004-512234).

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Hasson, A., Onshuus, A. & Peterzil, Y. Definable one dimensional structures in o-minimal theories. Isr. J. Math. 179, 297–361 (2010). https://doi.org/10.1007/s11856-010-0084-z

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  • DOI: https://doi.org/10.1007/s11856-010-0084-z

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