On local definability of holomorphic functions

Gareth Jones, Jonathan Kirby, Olivier le Gal, Tamara Servi

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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 that 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.

Original languageEnglish
Pages (from-to)1305-1326
Number of pages22
JournalThe Quarterly Journal of Mathematics
Volume70
Issue number4
Early online date16 Jul 2019
DOIs
Publication statusPublished - Dec 2019

Keywords

  • math.LO
  • 03C64, 14P10

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