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Abstract
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 nonzero 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 language  English 

Pages (fromto)  831870 
Number of pages  40 
Journal  Mathematics of computation 
Volume  90 
Issue number  328 
Early online date  27 Oct 2020 
DOIs  
Publication status  Published  Mar 2021 
Keywords
 Newman polynomials
 Littlewood polynomials
 complex Pisot numbers
 zero location
 unit disk

8th meeting of Young Lithuanian Mathematicians
Jonas Jankauskas (Organiser)
27 Dec 2019Activity: Participating in or organising an event › Participation in conference

Kevin Hare
Jonas Jankauskas (Host)
3 Feb 2019 → 16 Feb 2019Activity: Hosting a visitor › Hosting an academic visitor