Control of radii of convergence and extension of subanalytic functions

Let g: U\to \mathbb{R} denote a real analytic function on an open subset U of \mathbb{R}^n, and let \Sigma \subset \partial U denote the points where g does not admit a local analytic extension. We show that if g is semialgebraic (respectively, globally subanalytic), then \Sigma is semialgebraic (re...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2004-04, Vol.132 (4), p.997-1003
Main Author: Bierstone, Edward
Format: Article
Language:English
Subjects:
Algebra
Analytic functions
Exact sciences and technology
Functions of a complex variable
General mathematics
General, history and biography
Mathematical analysis
Mathematical functions
Mathematical theorems
Mathematics
Power series
Radii of convergence
Real functions
Sciences and techniques of general use
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Summary:Let g: U\to \mathbb{R} denote a real analytic function on an open subset U of \mathbb{R}^n, and let \Sigma \subset \partial U denote the points where g does not admit a local analytic extension. We show that if g is semialgebraic (respectively, globally subanalytic), then \Sigma is semialgebraic (respectively, subanalytic) and g extends to a semialgebraic (respectively, subanalytic) neighbourhood of \overline{U}\backslash\Sigma. (In the general subanalytic case, \Sigma is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series G centred at points b in the image of an analytic mapping \varphi, in terms of the radii of convergence of G\circ\widehat{\varphi}_a at points a\in\varphi^{-1}(b), where \widehat{\varphi}_a denotes the Taylor expansion of \varphi at a.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-03-07191-0