Control of radii of convergence and extension of subanalytic functions

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 $a$.