Series misdemeanors

TL;DR

This paper discusses the challenges and solutions in implementing and using truncated Puiseux series and related hierarchical series in computer algebra systems, highlighting improvements and limitations.

Contribution

It provides new ideas for designing more correct, efficient, and user-friendly implementations of Puiseux and hierarchical series in computer algebra systems.

Findings

Identifies key implementation issues in truncated Puiseux series

Proposes ideas for more reliable and efficient algorithms

Highlights limitations for users and developers

Abstract

Puiseux series are power series in which the exponents can be fractional and/or negative rational numbers. Several computer algebra systems have one or more built-in or loadable functions for computing truncated Puiseux series -- perhaps generalized to allow coefficients containing functions of the series variable that are dominated by any power of that variable, such as logarithms and nested logarithms of the series variable. Some computer-algebra systems also offer functions that can compute more-general truncated recursive hierarchical series. However, for all of these kinds of truncated series there are important implementation details that haven't been addressed before in the published literature and in current implementations. For implementers this article contains ideas for designing more convenient, correct, and efficient implementations or improving existing ones. For users, this article is a warning about some of these limitations. Many of the ideas in this article have been implemented in the computer-algebra within the TI-Nspire calculator, Windows and Macintosh products.