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Control of radii of convergence and extension of subanalytic functions


Author: Edward Bierstone
Journal: Proc. Amer. Math. Soc. 132 (2004), 997-1003
MSC (2000): Primary 13J07, 14P10, 32B20; Secondary 13J05, 32A10
DOI: https://doi.org/10.1090/S0002-9939-03-07191-0
Published electronically: September 5, 2003
MathSciNet review: 2045414
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Abstract: 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 $a$.


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Additional Information

Edward Bierstone
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: bierston@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07191-0
Received by editor(s): December 16, 2002
Published electronically: September 5, 2003
Additional Notes: The author’s research was partially supported by NSERC grant 0GP0009070
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2003 American Mathematical Society

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