On the existence of Newman and Littlewood multiples for certain integer polynomials

Abstract

Newman polynomials have coefficients in {0, 1} and constant term 1, whereas Littlewood polynomials have coefficients in {-1, 1}. We study two questions concerning the existence of Newman and Littlewood multiples for certain integer polynomials. For Newman multiples, we revisit a question of Hare and Mossinghoff [6]: whether there exists a real number sigma > 1 such that every P in Z[x] with no nonnegative real root and Mahler measure less than sigma has a Newman multiple. To bound any possible value of sigma from above, we seek polynomials with no nonnegative real root and no Newman multiple, and with Mahler measure as small as possible. Using the updated database of known small-Mahler-measure polynomials up to degree 200 [11], we test, for each listed polynomial p(x), both p(x) and its sign transform p(-x). These two polynomials have the same Mahler measure, but the existence of a Newman multiple is not invariant under the substitution x -> x. After a degree-bounded prefilter formulated as a mixed-integer linear optimization problem, we apply the Hare-Mossinghoff certification procedure to the remaining candidates. This yields 14 irreducible polynomials of Mahler measure less than 1.3, with no nonnegative real root and no Newman multiple. The smallest of their Mahler measures is 1.263095875491..., showing that any such sigma is at most this value. For Littlewood multiples, we return to the three Newman polynomials listed in Table 3 of Drungilas, Jankauskas, Junevicius, Klebonas and Siurys [3], for which the existence of a Littlewood multiple of smallest possible degree was left unresolved. We show that one of them attains this degree, whereas the other two have no Littlewood multiple at either of their first two possible degrees.