An efficient method of computing power expansions of algebraic functions is the method of Kung and Traub and is based on exact arithmetic. This paper shows a numeric approach is both feasible and accurate while also introducing a performance improvement to Kung and Traub's method based on the ramification extent of the expansions. A new method is then described for computing radii of convergence using a series comparison test. Series accuracies are then fitted to a simple log-linear function in their domain of convergence and found to have low variance. Algebraic functions up to degree 50 were analyzed and timed. A consequence of this work provided a simple method of computing the Riemann surface genus and was used as a cycle check-sum. Mathematica ver. 13.2 was used to acquire and analyze the data on a 4.0 GHz quad-core desktop computer.

翻译：计算代数函数幂展开的一种高效方法是基于精确算术的Kung-Traub方法。本文证明数值方法既可行又精确，同时基于展开的分支程度对Kung-Traub方法引入了一项性能改进。随后描述了一种使用级数比较检验计算收敛半径的新方法。在收敛域内对级数精度进行拟合，发现其符合简单的对数线性函数且方差较低。对阶数高达50的代数函数进行了分析与计时。本工作的一项成果是提供了一种计算Riemann曲面亏格的简便方法，并用作循环校验和。数据采集与分析在配备4.0 GHz四核台式计算机的Mathematica 13.2环境中完成。