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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Control of radii of convergence and extension of subanalytic functions
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by Edward Bierstone PDF
Proc. Amer. Math. Soc. 132 (2004), 997-1003 Request permission

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
  • 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
  • © Copyright 2003 American Mathematical Society
  • 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
  • MathSciNet review: 2045414