Given o-minimal expansion of semialgebraic geometry, Wilkie has studied the class of holomorphic functions which are (almost everywhere, locally) definable in this structure. Fixing such an o-minimal structure, a general problem is to determine which complex algebraic differential equations admit (nontrivial) solutions in this class and to compute them if there are any.

Taking “semialgebraic geometry” as our o-minimal structure, one recovers the classical problem which amounts to classifying the algebraic solutions of algebraic differential equations which is the subject of many fundamental conjectures. In my talk, I will describe this problem for the class of exponentially algebraic functions obtained by expanding semialgebraic geometry with the restricted complex exponential. The techniques use a combination of differential-algebraic methods going back to the work of Liouville on the problem of integration in finite terms and of model-theoretic methods about the blurred exponential.

This is joint work with Jonathan Kirby.