TY - JOUR

T1 - On Newman and Littlewood polynomials with prescribed number of zeros inside the unit disk

AU - Hare, Kevin

AU - Jankauskas, Jonas

PY - 2021/3

Y1 - 2021/3

N2 - 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.

AB - 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.

KW - Newman polynomials

KW - Littlewood polynomials

KW - complex Pisot numbers

KW - zero location

KW - unit disk

UR - http://www.scopus.com/inward/record.url?scp=85099611430&partnerID=8YFLogxK

U2 - 10.1090/MCOM/3570

DO - 10.1090/MCOM/3570

M3 - Article

VL - 90.2021

SP - 831

EP - 870

JO - Mathematics of computation

JF - Mathematics of computation

SN - 0025-5718

IS - March

ER -