Generic functions and quasiminimality

Abstract: One of the well-known accomplishments of model theory is the study of the field of complex numbers. It possesses numerous nice properties, including minimality, i.e. any definable subset is finite or cofinite. However, adding the exponential map to the structure makes it possible to define the ring of integers, preventing minimality and many other properties. Nevertheless, there is still hope that the complex exponential field is somewhat well-behaved. For instance, Zilber’s quasiminimality conjecture states that the complex exponential field is quasiminimal, i.e. every definable subset is countable or co-countable. Analogous conjectures were made, replacing the exponential map with other analytical functions.

In this talk we look at the theory of a generic function, as introduced by Zilber in 2002. As the main result, we prove that adding an entire generic function to the complex field gives a quasiminimal structure, and, moreover, this structure is unique up to an isomorphism. Thus we obtain a non-trivial example of an entire function which keeps the complex field quasiminimal.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.