Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

Fractalized cyclotomic polynomials

For each prime power , we realize the classical cyclotomic polynomial as one of a collection of different polynomials in . We show that the new polynomials are similar to in many ways, including that their discriminants all have the form . We show also that the new polynomials are more complicated than in other ways, including that their complex roots are generally fractal in appearance.

     

Inverse cyclotomic polynomials

Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x)=(xn−1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n<561 all coefficients of Ψn(x) are ?1 in absolute value. We establish various properties of the coefficients of Ψn(x), especially focusing on the easiest non-trivial case where n is composed of 3 distinct odd primes.     

A survey on coefficients of cyclotomic polynomials

Cyclotomic polynomials play an important role in several areas of mathematics and their study has a very long history, which goes back at least to Gauss (1801). In particular, the properties of their coefficients have been intensively studied by several authors, and in the last 10 years there has been a burst of activity in this field of research. This concise survey attempts to collect the main results regarding the coefficients of the cyclotomic polynomials and to provide all the relevant references to their proofs.     

Explicit factorizations of generalized Dickson polynomials of order 2m via generalized cyclotomic polynomials over finite fields

We derive explicit factorizations of generalized cyclotomic polynomials of order $2^m$ and generalized Dickson polynomials of the first kind of order $2^m$ over finite field $F_q$.     

Coefficients of ternary cyclotomic polynomials

It is customary to define a cyclotomic polynomial Φn(x) to be ternary if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn(x) and M(p) be the maximum of A(pqr). In 1968, Sister Marion Beiter (1968, 1971) [3] and [4] conjectured that . In 2008, Yves Gallot and Pieter Moree (2009) [6] showed that the conjecture is false for every p≥11, and they proposed the Corrected Beiter conjecture: . Here we will give a sufficient condition for the Corrected Beiter conjecture and prove it when p=7.     

On the Littlewood cyclotomic polynomials

In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp₁(±x)Φp₂(±xp₁)?Φpr(±xp₁p₂?pr−1), where N=p₁p₂?pr and the pi are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree α²pβ−1 with odd prime p or separable polynomials of any odd degree.     

Explicit factors of generalized cyclotomic polynomials and generalized Dickson polynomials of order 2m3 over finite fields

In this paper, we derive explicit factorizations of generalized cyclotomic polynomials and generalized Dickson polynomials of the first kind of order $2^{m}3$, over finite field $F_q$.     

Values of coefficients of cyclotomic polynomials

Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have

{a(k,p^ln)∣n,k∈N}=Z.     

On testing the divisibility of lacunary polynomials by cyclotomic polynomials

An algorithm is described that determines whether a given polynomial with integer coefficients has a cyclotomic factor. The algorithm is intended to be used for sparse polynomials given as a sequence of coefficient-exponent pairs. A running analysis shows that, for a fixed number of nonzero terms, the algorithm runs in polynomial time.

     

Period polynomials for generalized cyclotomic periods

The theory of cyclotomic period polynomials is developed for general periods of an arbitrary modulus, extending known results for the Gauss periods of prime modulus. Primes dividing the discriminant of the period polynomial are investigated, as are those primes dividing values of the period polynomial.Author has NSF grant MCS-8101860     

On the coefficients of ternary cyclotomic polynomials

We study coefficients of ternary cyclotomic polynomials Φ_pqr(z)=∏_ρ(z−ρ), where p, q, and r are distinct odd primes and the product is taken over all primitive pqrth roots of unity ρ.     

Values of coefficients of cyclotomic polynomials II

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. In part I it was proved that , in the case when m is a prime power. In this paper we show that the result also holds true in the case when m is an arbitrary positive integer.     

Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

We give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also corresponds (in four different ways) to a single Pisot polynomial from a certain restricted class, the cyclogenic Pisot polynomials. We investigate properties of this class of Pisot polynomials.     

A higher order family for the simultaneous inclusion of multiple zeros of polynomials

Starting from a suitable fixed point relation, a new family of iterative methods for the simultaneous inclusion of multiple complex zeros in circular complex arithmetic is constructed. The order of convergence of the basic family is four. Using Newtons and Halleys corrections, we obtain families with improved convergence. Faster convergence of accelerated methods is attained with only few additional numerical operations, which provides a high computational efficiency of these methods. Convergence analysis of the presented methods and numerical results are given. AMS subject classification 65H05, 65G20, 30C15