On the cyclotomic polynomials with +1 or -1 coefficients
Shabnam Akhtari · Summit (Simon Fraser University) · 2004
In this thesis, 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 give a complete characterization of all cyclotomic polynomials with odd coefficients.They also prove that a polynomial p(x) with coefficients f 1 of even degree N -1 is cyclotomic if and only if where N = plp2.. .prand the pi are primes, not necessarily distinct.Here a,(%) := is the pth cyclotomic polynomial.Based on substantial computation, they also conjecture that this characterization also holds for polynomials of odd degree with f 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 2tpT -1 with odd prime p or separable polynomials of any odd degree.We also give a simpler proof of Borwein and Choi's result.A constant source of appreciation and respect is my family: My Dad, whose memory is still encouraging, my very suffering mom, my caring husband and my lovely sisters.