AI 中文总结
研究整系数多项式的纽曼倍数和利特伍德倍数存在性问题，通过利用小马勒测度多项式数据库及相关程序，得到马勒测度小于1.3且无纽曼倍数的多项式，还确定了三个纽曼多项式中利特伍德倍数的情况。
AI 中文摘要
纽曼多项式系数在{0, 1}且常数项为1，利特伍德多项式系数在{-1, 1}。我们研究关于某些整系数多项式的纽曼倍数和利特伍德倍数存在性的两个问题。对于纽曼倍数，重新审视Hare和Mossinghoff的问题：是否存在实数σ>1，使得每个无非负实根且马勒测度小于σ的整系数多项式都有纽曼倍数。为从上方界定σ的可能值，寻找无纽曼倍数且马勒测度尽可能小的多项式。利用已知小马勒测度多项式数据库测试，经预筛选和认证程序，得到14个马勒测度小于1.3的不可约多项式，最小为1.263095875491...。对于利特伍德倍数，研究Drungilas等人列出的三个纽曼多项式，表明其中一个达到最小可能次数，另两个在前两个可能次数都没有利特伍德倍数。
英文摘要
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.
