TY - JOUR

T1 - On the Littlewood cyclotomic polynomials

AU - Akhtari, Shabnam

AU - Choi, Stephen K.

N1 - Funding Information:
* Corresponding author. E-mail addresses: akhtari@math.ubc.ca (S. Akhtari), kkchoi@cecm.sfu.ca (S.K. Choi). 1 Research of Stephen Choi was supported by NSERC of Canada.

PY - 2008/4

Y1 - 2008/4

N2 - 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 (± xp1 p2 ⋯ pr - 1), where N = p1 p2 ⋯ pr and the pi are primes, not necessarily distinct. Here Φp (x) : = frac(xp - 1, x - 1) 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 2α pβ - 1 with odd prime p or separable polynomials of any odd degree.

AB - 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 (± xp1 p2 ⋯ pr - 1), where N = p1 p2 ⋯ pr and the pi are primes, not necessarily distinct. Here Φp (x) : = frac(xp - 1, x - 1) 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 2α pβ - 1 with odd prime p or separable polynomials of any odd degree.

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U2 - 10.1016/j.jnt.2007.02.010

DO - 10.1016/j.jnt.2007.02.010

M3 - Article

AN - SCOPUS:39749119320

SN - 0022-314X

VL - 128

SP - 884

EP - 894

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 4

ER -