Representation, simplification and display of fractional powers of rational numbers in computer algebra

TL;DR

This paper examines the simplification of fractional powers of positive rational numbers in computer algebra, highlighting existing issues and proposing methods to improve efficiency and correctness without relying on costly integer factorization.

Contribution

It identifies deficiencies in current algorithms and discusses alternative forms and techniques to enhance simplification of rational powers in computer algebra systems.

Findings

Current algorithms can be inefficient for large numbers.

Existing systems may produce incorrect or bulky results.

Proposed methods avoid costly integer factorization.

Abstract

Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail. Since they are such a restricted subset of algebraic numbers, it might seem that good simplification of them must already be implemented in all widely used computer algebra systems. However, the algorithm taught in beginning algebra uses integer factorization, which can consume unacceptable time for the large numbers that often arise within computer algebra. Therefore some systems apparently use various ad hoc techniques that can return an incorrect result because of not simplifying to 0 the difference between two equivalent such expressions. Even systems that avoid this flaw often do not return the same result for all equivalent such input forms, or return an unnecessarily bulky result that does not have any other compensating useful property. This article identifies some of these deficiencies, then describes the advantages and disadvantages of various alternative forms and how to overcome the deficiencies without costly integer factorization.