Submission date: 16 March 2010

Abstract:

Let M be an o-minimal expansion of the field of real numbers that defines nontrivial arcs of both the sine and exponential functions. Let G be a collection of images of solutions on intervals to differential equations y'=F(y), where F ranges over all R-linear transformations R^n --> R^n and n ranges over all positive integers. Then either the expansion of M by the elements of G is as well behaved relative to M as one could reasonably hope for, or it defines the set of all integers Z, and thus is as complicated as possible. In particular, if M defines any irrational power functions, then the expansion of M by the elements of G either is o-minimal or defines Z.

Mathematics Subject Classification: Primary 03C64; Secondary 34A30

Keywords and phrases: expansion of the real field, o-minimal, linear vector field

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