极紧集外解析函数的“最大”亚纯定义域的存在性
Existence of a "maximal" domain of meromorphy for an analytic function outside a polar compact set
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中文总结 AI 辅助
本文指出Stahl关于极紧集外解析芽的最小对数容量紧集存在性的证明存在关键错误，采用全新思路给出了该结论的正确证明，并探讨了更一般的Stahl猜想。
中文摘要 AI 辅助
设E是复平面$\u2102$中的一个极紧集。设$f_\infty$是无穷远点处的一个芽，它可以沿着位于黎曼球面$\widehat{\mathbb C}\setminus E$中、以无穷远点为起点的任意路径$\gamma$进行解析延拓。1985至1986年间，Herbert Stahl给出了一个关于由此类芽$f_\infty$构造的对角Padé逼近收敛性的基本定理的证明，此后该定理便以他的名字命名。在他的证明中，一个关键的支撑结论是：在所有满足“芽$f_\infty$可作为单值亚纯函数延拓到$\widehat{\mathbb C}\setminus K$”的紧集K中，存在一个对数容量最小的紧集$S_{f_\infty}$。遗憾的是，H. Stahl在1985年给出的这一结论的证明存在一处关键错误。令人惊讶的是，这一错误出现在Stahl1985年的原始论文中，又在他2012年的最后一篇预印本中被重复，且据我们所知，此前从未有人指出过这一问题！在本文中，我们阐释了Stahl的这一严重错误，并给出了紧集$S_{f_\infty}$存在性的正确证明。我们强调，我们的证明采用了与Stahl的思路完全不同的想法。此外，我们还讨论了更具一般性的Stahl猜想：对于任意芽$f_\infty$，在不对芽$f_\infty$可延拓的路径做任何假设的情况下，紧集$S_{f_\infty}$仍然存在。
英文摘要
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 $γ$ 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 Padé 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.
