This talk will regard some first-order properties of ordered groups of germs of functions that are definable in an o-minimal structure. I will describe my progress toward finding a tame first-order theory of some of these groups, and give some elements of valuation theory on these ordered groups.

In 2002 Zilber introduced the theory of a generic function on a field [4], coinciding with the limit theory of generic polynomials from [2]. Axiomatized in first-order logic by a version of Schanuel property and existential closedness, this theory turns out to be ω-stable. As shown by Wilkie [3] and Koiran [1], one can explicitly construct such a generic function on the complex plane in a form of a Taylor series, using the ideas behind Liouville numbers.

In this talk we look further into the properties of the theory of generic functions. As the main result, we show that adding any of these generic functions to the complex field gives an isomorphic structure, which ought to be quasiminimal, i.e. any definable subset has to be countable or cocountable. Thus we obtain a non-trivial example of an entire function which keeps the complex field quasiminimal.

Bibliography

1. Pascal Koiran. The theory of liouville functions. Journal of Symbolic Logic, 68(2):353–365, 2003.
2. Pascal Koiran. The limit theory of generic polynomials, pages 242–254. Lecture Notes In Logic,. 03 2005.
3. A. J. Wilkie. Liouville functions, page 383–391. Lecture Notes in Logic. Cambridge University Press, 2005.
4. Boris Zilber. A theory of a generic function with derivations, page 85–99. Contemporary mathematics. American Mathematical Society, 2002.

In this talk, I will discuss joint work with P. Speissegger in which we prove that the \(\Gamma\) function and Riemann’s \(\zeta\) function are o-minimal on certain unbounded complex domains. I will also discuss some applications of this result.