A numeric study of power expansions around singular points of algebraic functions, their radii of convergence, and accuracy profiles

Abstract

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.

28 pages, 19 tables, 12 figures Ver 2 corrections: (1) Added absolute value bars for note about R in summary report of Test Case 6 (2) 5-degree function f_2 in eqn. (24) was wrong function for this test case. Changed to correct 4-degree function. All tables of this test case reflected the results of the 4-degree function and were not changed