2019/10/30 by Kevin G. Hare, Hare, Kevin G., Jonas Jankauskas +1
Mathematics · #11B83 #11R06 #11R09 #11Y99 #12D10 #26C10 #30C15 #65H04 #93A99 #Advanced Mathematical Identities #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1910.13994
openalex publication_date 2019/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study 0, 1 and -1, 1 polynomials f(z), called Newman and\nLittlewood polynomials, that have a prescribed number N(f) of zeros in the\nopen unit disk \D = z \∈ \ℂ: |z| < 1 . For every pair\n(k, n) \∈ \ℕ2, where n \≥ 7 and k \∈ [3, n-3], we prove that\nit is possible to find a 0, 1 --polynomial f(z) of degree deg\nf=n with non--zero constant term f(0) \≠ 0, such that N(f)=k and f(z)\n\≠ 0 on the unit circle \∂\D. On the way to this goal, we\nanswer a question of D.~W.~Boyd from 1986 on the smallest degree Newman\npolynomial that satisfies |f(z)| > 2 on the unit circle \∂\n\D. This polynomial is of degree 38 and we use this special\npolynomial in our constructions. We also identify (without a proof) all\nexceptional (k, n) with k \∈ 1, 2, 3, n-3, n-2, n-1 , for which no such\n 0, 1 --polynomial of degree n exists: such pairs are related to regular\n(real and complex) Pisot numbers.\n Similar, but less complete results for -1, 1 polynomials are\nestablished. We also look at the products of spaced Newman polynomials and\nconsider the rotated large Littlewood polynomials. Lastly, based on our data,\nwe formulate a natural conjecture about the statistical distribution of N(f)\nin the set of Newman and Littlewood polynomials.\n