Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, $${f \in \mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$f∈A(C¯\A), $${\# A< \infty}$$#A<∞. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Padé approximants for f. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single-valued branch. Thus the domain of convergence corresponds to the maximal (in the sense of minimal boundary) domain of single-valued holomorphy for the analytic function $${f\in\mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$f∈A(C¯\A). The complete proof of Nuttall’s conjecture (even in a more general setting where the set A has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive strong asymptotics for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that A is a finite set of branch points of f which have the algebro-logarithmic character and which are placed in a generic position. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.

中文翻译：

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具有分支点的函数的 Padé 近似——Nuttall-Stahl 多项式的强渐近

设 f 是无穷远解析函数的胚芽，该函数可以在没有有限点集的复平面中沿任何路径解析地延续，$${f \in \mathcal{A}(\bar{\mathbb{C }} \setminus A)}$$f∈A(C¯\A), $${\# A< \infty}$$#A<∞。J. Nuttall 提出了函数具有单值分支的 f 的最大域与 f 的对角线 Padé 近似的收敛域之间的重要关系。Padé 逼近是有理函数，因此是单值的，在它们的收敛域中逼近 f 的全纯分支。同时，他们的大多数极点趋向于收敛域的边界，并且他们的极限分布模型的支持使函数 f 成为单值的切割系统。Nuttall 已经推测（并在许多重要的特殊情况下证明了），在所有其他将函数 f 转换为单值分支的系统中，这种切割系统具有最小的对数容量。因此收敛域对应于解析函数 $${f\in\mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$f∈A(C¯\A)。纳托尔猜想的完整证明（即使在集合 A 的对数容量为 0 的更一般设置中）由 H. Stahl 获得。在这项工作中，我们在相当一般的环境中为这个问题的对角线 Padé 近似的分母推导出强渐近性。我们假设 A 是 f 的有限分支点集，这些分支点具有代数对数特征并且位于一般位置。