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On local definability of holomorphic functions

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On local definability of holomorphic functions. / Jones, Gareth; Kirby, Jonathan; le Gal, Olivier; Servi, Tamara.

In: The Quarterly Journal of Mathematics, Vol. 70, No. 4, 12.2019, p. 1305-1326.

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Jones, Gareth ; Kirby, Jonathan ; le Gal, Olivier ; Servi, Tamara. / On local definability of holomorphic functions. In: The Quarterly Journal of Mathematics. 2019 ; Vol. 70, No. 4. pp. 1305-1326.

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@article{ac1f29340253455d9540f88aaed0f952,
title = "On local definability of holomorphic functions",
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.",
keywords = "math.LO, 03C64, 14P10",
author = "Gareth Jones and Jonathan Kirby and {le Gal}, Olivier and Tamara Servi",
year = "2019",
month = dec,
doi = "10.1093/qmath/haz015",
language = "English",
volume = "70",
pages = "1305--1326",
journal = "The Quarterly Journal of Mathematics",
issn = "0033-5606",
publisher = "Oxford University Press",
number = "4",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - On local definability of holomorphic functions

AU - Jones, Gareth

AU - Kirby, Jonathan

AU - le Gal, Olivier

AU - Servi, Tamara

PY - 2019/12

Y1 - 2019/12

N2 - 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.

AB - 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.

KW - math.LO

KW - 03C64, 14P10

UR - http://www.scopus.com/inward/record.url?scp=85082451133&partnerID=8YFLogxK

U2 - 10.1093/qmath/haz015

DO - 10.1093/qmath/haz015

M3 - Article

VL - 70

SP - 1305

EP - 1326

JO - The Quarterly Journal of Mathematics

JF - The Quarterly Journal of Mathematics

SN - 0033-5606

IS - 4

ER -

ID: 129743665