Cyclotomic Factors and LRS-Degeneracy
John Abbott, Nico Mexis
TL;DR
This work delivers three practical algorithms for polynomials in ${\mathbb Z}[x]$: (i) a fast cyclotomicity test that also determines the index, (ii) a method to identify all cyclotomic factors with low false-positive risk, and (iii) an improved, modular approach to detect and order LRS-degeneracy. The methods blend quick prechecks, prefix-based analyses, evaluation strategies, and modular arithmetic to prioritize efficiency over asymptotic analysis, with all algorithms implemented in CoCoALib. The results show substantial practical speedups for LRS-degeneracy detection compared with CDM11, while providing robust cyclotomic recognition and factor-detection capabilities. These tools have direct utility in number theory computations and related algorithmic tasks, and they emphasize a pragmatic, implementable approach to otherwise intricate algebraic problems.
Abstract
We present three new, practical algorithms for polynomials in $\mathbb{Z}[x]$: one to test if a polynomial is cyclotomic, one to determine which cyclotomic polynomials are factors, and one to determine whether the given polynomial is LRS-degenerate. A polynomial is "LRS-degenerate" iff it has two distinct roots $α, β$ such that $β= ζα$ for some root of unity $ζ$. All three algorithms are based on "intelligent brute force". The first two produce the indexes of the cyclotomic polynomials; the third produces a list of degeneracy orders. The algorithms are implemented in CoCoALib.