Control of radii of convergence and extension of subanalytic functions

Abstract

Let g denote a real analytic function on an open subset U of Euclidean space, and let S denote the boundary points of U where g does not admit a local analytic extension. We show that if g is semialgebraic (respectively, globally subanalytic), then S is semialgebraic (respectively, subanalytic) and g extends to a neighbourhood of cl(U)\S as an analytic function that is semialgebraic (respectively, globally subanalytic). (In the general subanalytic case, S is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series G centred at points in the image of an analytic mapping, in terms of the radii of convergence of the pull-backs of G at points of the source.