## 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 $a$.## References

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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