The Power of Vocabulary: The Case of Cyclotomic Polynomials

TL;DR

This paper explores how the choice of vocabulary, such as cyclotomic polynomials, impacts the computational complexity in computer algebra, proposing extended vocabularies to improve efficiency.

Contribution

It formalizes the influence of vocabulary on problem complexity and suggests practical extensions using cyclotomic polynomials and $x^n-1$ for better computational performance.

Findings

Vocabulary choice significantly affects computational complexity.

Extending vocabulary with cyclotomic polynomials improves efficiency.

Using both cyclotomic and $x^n-1$ options offers practical benefits.

Abstract

We observe that the vocabulary used to construct the "answer" to problems in computer algebra can have a dramatic effect on the computational complexity of solving that problem. We recall a formalization of this observation and explain the classic example of sparse polynomial arithmetic. For this case, we show that it is possible to extend the vocabulary so as reap the benefits of conciseness whilst avoiding the obvious pitfall of repeating the problem statement as the "solution". It is possible to extend the vocabulary either by irreducible cyclotomics or by $x^n-1$: we look at the options and suggest that the pragmatist might opt for both.