On Newman and Littlewood polynomials with prescribed number of zeros inside the unit disk

Kevin Hare, Jonas Jankauskas

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We study {0,1} and {−1,1} polynomials f(z), called Newman and Littlewood polynomials, that have a prescribed number N(f) of zeros in the open unit disk D={z∈C:|z|<1}. For every pair (k,n)∈N^2, where n≥7 and k∈[3,n−3], we prove that it is possible to find a {0,1}--polynomial f(z) of degree deg f=n with non--zero constant term f(0)≠0, such that N(f)=k and f(z)≠0 on the unit circle ∂D. On the way to this goal, we answer a question of D. W. Boyd from 1986 on the smallest degree Newman polynomial that satisfies |f(z)|>2 on the unit circle ∂D. This polynomial is of degree 38 and we use this special polynomial in our constructions. We also identify (without a proof) all exceptional (k,n) with k∈{1,2,3,n−3,n−2,n−1}, for which no such {0,1}--polynomial of degree n exists: such pairs are related to regular (real and complex) Pisot numbers. Similar, but less complete results for {−1,1} polynomials are established. We also look at the products of spaced Newman polynomials and consider the rotated large Littlewood polynomials. Lastly, based on our data, we formulate a natural conjecture about the statistical distribution of N(f) in the set of Newman and Littlewood polynomials.
Original languageEnglish
Pages (from-to)831-870
Number of pages40
JournalMathematics of computation
Issue numberMarch
Early online date27 Oct 2020
Publication statusPublished - Mar 2021


  • Newman polynomials
  • Littlewood polynomials
  • complex Pisot numbers
  • zero location
  • unit disk
  • 8th meeting of Young Lithuanian Mathematicians

    Jonas Jankauskas (Organiser)

    27 Dec 2019

    Activity: Participating in or organising an eventParticipation in conference

  • Kevin Hare

    Jonas Jankauskas (Host)

    3 Feb 201916 Feb 2019

    Activity: Hosting a visitorHosting an academic visitor

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