Factoring Variants of Chebyshev Polynomials with Minimal Polynomials of $\cos(\frac{2π}{d})$

Abstract

We solve the problem of factoring polynomials Vn(x)±1 and Wn(x)±1 where Vn(x) and Wn(x) are Chebyshev polynomials of the third and fourth kinds. The method of proof is based on previous work by Wolfram [12] for factoring variants of Chebyshev polynomials of the first and second kinds, Tn(x) ± 1 and Un(x) ± 1. We also show that, in general, there are no factorizations of variants of Chebyshev polynomials of the fifth and sixth kinds, Xn(x) ± 1 and Yn(x) ± 1 using minimal polynomials of cos( 2π d ).