On Newman polynomials which divide no Littlewood polynomial

Abstract

Recall that a polynomial P(x) G Z[x] with coefficients 0, 1 and constant term 1 is called a Newman polynomial, whereas a polynomial with coefficients -1, 1 is called a Littlewood polynomial. Is there an algebraic number ol which is a root of some Newman polynomial but is not a root of any Littlewood polynomial? In other words (but not equivalently), is there a Newman polynomial which divides no Littlewood polynomial? In this paper, for each Newman polynomial P of degree at most 8, we find a Littlewood polynomial divisible by P. Moreover, it is shown that every trinomial l + uxa +vxb, where a < b are positive integers and u, v € {1, 1}, so, in particular, every Newman trinomial 1 + xa + xb, divides some Littlewood polynomial. Nevertheless, we prove that there exist Newman polynomials which divide no Littlewood polynomial, e.g., x9+x6+x2+x+l. This example settles the problem 006:07 posed by the first named author at the 2006 West Coast Number Theory conference. It also shows that the sets of roots of Newman polynomials Vjv» Littlewood polynomials Vc and {-1,0,1} polynomials V are distinct in the sense that between them there are only trivial relations V/v C V and Vc C V. Moreover, V jz Vc U Vjsf. The proofs of several main results (after some preparation) are computational.