Binary Cyclotomic Polynomials: Representation via Words and Algorithms

TL;DR

This paper introduces an efficient linear-time algorithm to represent binary cyclotomic polynomials as words over a ternary alphabet, enabling better understanding and extension of their properties.

Contribution

It develops a compact, linear-time algorithm for representing binary cyclotomic polynomial coefficients as words, extending to numerical semi-groups of dimension 2.

Findings

Algorithm computes compact representations efficiently

Reveals properties of polynomial coefficients

Extends to numerical semi-groups of dimension 2

Abstract

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees the vector of coefficients of the polynomial as a word on a ternary alphabet $\{-1,0 ,+1\}$. It designs an efficient algorithm that computes a compact representation of this word. This algorithm is of linear time with respect to the size of the output, and, thus, optimal. This approach allows to recover known properties of coefficients of binary cyclotomic polynomials, and extends to the case of polynomials associated with numerical semi-groups of dimension 2.