Existence of a "maximal" domain of meromorphy for an analytic function outside a polar compact set

Abstract

Let E be a polar compact set in $\mathbb C$. Let $f_\infty$ be a germ at $\infty$ that can be analytically continued along an arbitrary path $\gamma$ lying in $\widehat{\mathbb C}\setminus E$ and starting at the point $\infty$. In 1985--1986 Herbert Stahl presented his proof of a fundamental theorem on the convergence of diagonal Pad\'e approximants constructed from such germ $f_\infty$. Since then, this theorem has borne his name. A crucial role in his proof is played by the existence of a compact set $S_{f_\infty}$ of minimal logarithmic capacity among all compact sets $K$ such that the germ $f_\infty$ extends as a single-valued meromorphic function to $\widehat{\mathbb C}\setminus K$. Unfortunately, the proof of this fact presented by H. Stahl in 1985 contains a crucial mistake. It is surprising that this mistake was made in the original Stahl's paper in 1985 and repeated in his last preprint in 2012, and, as far as we know, no one has pointed it out before! In this paper we explain this serious Stahl's mistake and present a correct proof of the existence of a compact set $S_{f_\infty}$. We emphasize that our proof will use ideas completely different from Stahl's ideas. Also we discuss the more general Stahl's conjecture about the existence of a compact set $S_{f_\infty}$ for an arbitrary germ $f_\infty$ without any assumption on the paths along that the germ $f_\infty$ can be continued.