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 dimM(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.
Bibliographical noteFunding Information:
∗ Supported by the EPSRC grant no. EP C52800X 1. ∗∗Partially supported by the EC FP6 through the Marie Curie Network MODNET (MRTN-CT-2004-512234). Received April 14, 2007 and in revised form January 21, 2009
ASJC Scopus subject areas
- Mathematics (all)