Calculating cyclotomic polynomials of very large height
Andrew Arnold, Michael Monagan · 2008
We present two algorithms to calculate n(z), the nth cyclotomic polynomial. The rst algorithm calculates n(z) by a series of polynomial divisions, which we do using the discrete Fourier transform. The second algorithm calculates n(z) as a quotients of products of sparse power series. These algorithms, described in detail in the paper, were used to calculate cyclotomic polynomials of large height and length. In particular, we have found cyclotomic polynomials n(z) of minimal order n whose height is greater than n, n 2 , n 3 , and n 4 , respectively. We include these results as well as other examples of cyclotomic polynomials of unusually large height, and bounds on the kth coecient for all cyclotomic polynomials.