## 2012.5: Some local definability theory for holomorphic functions

2012.5:
A J Wilkie
(2008)
*Some local definability theory for holomorphic functions.*
In:
Model Theory with Applications to Algebra and Analysis.
London Mathematical Society Lecture Note Series, 1 (349).
Cambridge University Press, Cambridge, pp. 197-213.
ISBN 978-0-521-70908-8

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## Abstract

Let F be a collection of holomorphic functions and let R(PR(F)) denote the reduct of the structure Ran to the ordered field operations together with the set of proper restrictions (see below) of the real and imaginary parts of all functions in F. We ask the question: Which holomorphic functions are locally definable (ie have their real and imaginary parts locally definable) in the structure R(PR(F))? It is easy to see that the collection of all such functions is closed under composition, partial differentiation, implicit definability (via the Implicit Function Theorem in one dependent variable) and Schwarz Reflection. We conjecture that this exhausts the possibilities and we prove as much in the neighbourhood of generic points. More precisely, we show that these four operations determine the natural pregeometry associated with R(PR(F))-definable, holomorphic functions.

Item Type: | Book Section |
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Uncontrolled Keywords: | Model theory, complex analysis |

Subjects: | MSC 2000 > 03 Mathematical logic and foundations MSC 2000 > 30 Functions of a complex variable |

MIMS number: | 2012.5 |

Deposited By: | Prof Alex J Wilkie |

Deposited On: | 05 January 2012 |

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