Calculating cyclotomic polynomials

Abstract. We present three algorithms to calculate Φn(z), the nth cyclotomic polynomial. The first algorithm calculates Φn(z) by a series of polynomial divisions, which we perform using the fast Fourier transform. The second algorithm calculates Φn(z) as a quotient of products of sparse power series. These two algorithms, described in detail in the paper, were used to calculate cyclotomic polynomials of large height and length. In particular, we have found the least n for which the height of Φn(z) is greater than n, n2, n3, and n4, respectively. The third algorithm, the big prime algorithm, generates the terms of Φn(z) sequentially, in a manner which reduces the memory cost. We use the big prime algorithm to find the minimal known height of cyclotomic polynomials of order five. We include these results as well as other examples of cyclotomic polynomials of unusually large height, and bounds on the coefficient of the term of degree k for all cyclotomic polynomials.

• Type: Journal
• Year: 2011
• Where: MOC
• Authors: Andrew Arnold, Michael B. Monagan
• Keywords: Computer Networks, Cyclotomic Polynomials, MOC 2011, Polynomial, Prime Algorithm