📝 SummarXiv

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 $\alpha, \beta$ such that $\beta = \zeta \alpha$ for some root of unity $\zeta$. 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.