## Control of radii of convergence and extension of subanalytic functions

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- by Edward Bierstone
- Proc. Amer. Math. Soc.
**132**(2004), 997-1003 - DOI: https://doi.org/10.1090/S0002-9939-03-07191-0
- Published electronically: September 5, 2003
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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 $a$.## References

- Edward Bierstone and Pierre D. Milman,
*Semianalytic and subanalytic sets*, Inst. Hautes Études Sci. Publ. Math.**67**(1988), 5–42. MR**972342**, DOI 10.1007/BF02699126 - Jacques Chaumat and Anne-Marie Chollet,
*On composite formal power series*, Trans. Amer. Math. Soc.**353**(2001), no. 4, 1691–1703 (English, with English and French summaries). MR**1806723**, DOI 10.1090/S0002-9947-01-02733-7 - Heisuke Hironaka,
*Triangulations of algebraic sets*, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 165–185. MR**0374131** - A. Mouze,
*Sur la composition de séries formelles à croissance contrôlée*, Ann. Scuola Norm. Sup. Pisa Cl. Sci (5)**1**(2002), 73–92. - Jean-Claude Tougeron,
*Sur les racines d’un polynôme à coefficients séries formelles*, Real analytic and algebraic geometry (Trento, 1988) Lecture Notes in Math., vol. 1420, Springer, Berlin, 1990, pp. 325–363 (French). MR**1051220**, DOI 10.1007/BFb0083927

## Bibliographic 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