On local definability of holomorphic functions

TL;DR

This paper investigates the local definability of holomorphic functions from a given collection, revealing that Wilkie's existing description is incomplete near non-generic points and proposing the need for additional operations.

Contribution

It demonstrates that Wilkie's description of local definability is insufficient at non-generic points and introduces new examples suggesting necessary extensions.

Findings

Wilkie's description is incomplete near non-generic points

Three new holomorphic functions indicate the need for additional operations

Interaction between resolution of singularities and definability is illustrated

Abstract

Given a collection A of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from A. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from A in the neighbourhood of a generic point. We prove that this description is no longer complete in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions which suggest that at least three new operations need to be added to Wilkie's description in order to capture local definability in its entirety. The constructions illustrate the interaction between resolution of singularities and definability in the o-minimal setting.