Expansions of o-minimal structures on the real field by trajectories of linear vector fields

Abstract

Let ℜ 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 N. Then either the expansion of ℜ by the elements of g is as well behaved relative to ℜ 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 ℜ defines any irrational power functions, then the expansion of ℜ by the elements of g either is o-minimal or defines Z.