2013/02/08 by Albert D. Rich, Rich, Albert D., David R. Stoutemyer +1
Computer Science · #11 Number Theory #Algebraic Number Theory Computations #FOS: Computer and information sciences #I.1.1 #Logic, programming, and type systems #Numerical Methods and Algorithms #Polynomial and algebraic computation #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.1302.2169
openalex publication_date 2013/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Simplification of fractional powers of positive rational numbers and of sums,\nproducts and powers of such numbers is taught in beginning algebra. Such\nnumbers can often be expressed in many ways, as this article discusses in some\ndetail. Since they are such a restricted subset of algebraic numbers, it might\nseem that good simplification of them must already be implemented in all widely\nused computer algebra systems. However, the algorithm taught in beginning\nalgebra uses integer factorization, which can consume unacceptable time for the\nlarge numbers that often arise within computer algebra. Therefore some systems\napparently use various ad hoc techniques that can return an incorrect result\nbecause of not simplifying to 0 the difference between two equivalent such\nexpressions. Even systems that avoid this flaw often do not return the same\nresult for all equivalent such input forms, or return an unnecessarily bulky\nresult that does not have any other compensating useful property. This article\nidentifies some of these deficiencies, then describes the advantages and\ndisadvantages of various alternative forms and how to overcome the deficiencies\nwithout costly integer factorization.\n