We investigate the numbers of complex zeros of Littlewood polynomials p(z) (polynomials with coefficients {−1, 1}) inside or on the unit circle |z| = 1, denoted by N(p) and U(p), respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for N(p), U(p) for polynomials p(z) of these types. We show that if n + 1 is a prime number, then for each integer k, 0 ≤ k ≤ n − 1, there exists a Littlewood polynomial p(z) of degree n with N(p) = k and U(p) = 0. Furthermore, we describe some cases where the ratios N(p)/n and U(p)/n have limits as n → ∞ and find the corresponding limit values.

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A Newman polynomial has all the coefficients in {0, 1} and constant term 1, whereas a Littlewood polynomial has all coefficients in {−1, 1}. We call P (X) ∈ Z[X] a Borwein polynomial if all its coefficients belong to {−1, 0, 1} and P (0) = 0. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle |z| = 1 has a non-zero multiple in Z[X] with coefficients in a finite set D ⊂ Z, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.

Abstract. Recall that a polynomial P (x) ∈ 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 α 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 1 + ux a + vx b , where a < b are positive integers and u, v ∈ {−1, 1}, so, in particular, every Newman trinomial 1 + x a + x b , divides some Littlewood polynomial. Nevertheless, we prove that there exist Newman polynomials which divide no Littlewood polynomial, e.g., x 9 +x 6 +x 2 +x+1. 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 V N , Littlewood polynomials V L and {−1, 0, 1} polynomials V are distinct in the sense that between them there are only trivial relationsThe proofs of several main results (after some preparation) are computational.

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