TY - JOUR

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

AU - Jankauskas, Jonas

AU - Hare, Kevin

PY - 2019/10/27

Y1 - 2019/10/27

N2 - We study f0; 1g and f1; 1g polynomials f(z), calledNewman and Littlewood polynomials, that have a prescribed num-ber N(f) of zeros in the open unit disk D = fz 2 C : jzj < 1g.For every pair (k; n) 2 N2, where n 7 and k 2 [3; n 3], weprove that it is possible to nd a f0; 1g{polynomial f(z) of de-gree deg f = n with non{zero constant term f(0) 6= 0, such thatN(f) = k and f(z) 6= 0 on the unit circle @D. On the way tothis goal, we answer a question of D. W. Boyd from 1986 on thesmallest degree Newman polynomial that satises jf(z)j > 2 onthe unit circle @D. This polynomial is of degree 38 and we use thisspecial polynomial in our constructions. We also identify (withouta proof) all exceptional (k; n) with k 2 f1; 2; 3; n3; n2; n1g,for which no such f0; 1g{polynomial of degree n exists: such pairsare related to regular (real and complex) Pisot numbers.Similar, but less complete results for f1; 1g polynomials areestablished. We also look at the products of spaced Newman poly-nomials and consider the rotated large Littlewood polynomials.Lastly, based on our data, we formulate a natural conjecture aboutthe statistical distribution of N(f) in the set of Newman and Lit-tlewood polynomials.

AB - We study f0; 1g and f1; 1g polynomials f(z), calledNewman and Littlewood polynomials, that have a prescribed num-ber N(f) of zeros in the open unit disk D = fz 2 C : jzj < 1g.For every pair (k; n) 2 N2, where n 7 and k 2 [3; n 3], weprove that it is possible to nd a f0; 1g{polynomial f(z) of de-gree deg f = n with non{zero constant term f(0) 6= 0, such thatN(f) = k and f(z) 6= 0 on the unit circle @D. On the way tothis goal, we answer a question of D. W. Boyd from 1986 on thesmallest degree Newman polynomial that satises jf(z)j > 2 onthe unit circle @D. This polynomial is of degree 38 and we use thisspecial polynomial in our constructions. We also identify (withouta proof) all exceptional (k; n) with k 2 f1; 2; 3; n3; n2; n1g,for which no such f0; 1g{polynomial of degree n exists: such pairsare related to regular (real and complex) Pisot numbers.Similar, but less complete results for f1; 1g polynomials areestablished. We also look at the products of spaced Newman poly-nomials and consider the rotated large Littlewood polynomials.Lastly, based on our data, we formulate a natural conjecture aboutthe statistical distribution of N(f) in the set of Newman and Lit-tlewood polynomials.

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 -