11/16/2019, Jonathan Kirby, University of East Anglia

Jonathan Kirby, University of East Anglia

Local Definability of Holomorphic Functions

Given a collection F of complex or real analytic functions, one can ask what other functions are obtainable from them by finitary algebraic operations. If we just mean polynomial operations we get some field of functions.
If we include as algebraic operations such things as taking implicit functions, maybe in several variables, we get a much more interesting framework, which is closely related to the theory of local definability in an o-minimal setting, starting with suitable restrictions of the functions in F.

O-minimality is a setting for tame topology of real- or complex-analytic functions which does not allow for “bad” singularities. However some more tame singularities can occur. In this talk I will explain work showing what singularities we have to consider to get a characterisation of the locally definable functions in terms of complex analytic operations.

Ax’s theorem on the differential algebra version of Schanuel’s conjecture is important to give one counterexample, and also for some applications to exponential and elliptic functions.

This is joint work with Gareth Jones, Olivier Le Gal, and Tamara Servi.