## Abstract

We study f0; 1g and f1; 1g polynomials f(z), called

Newman 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], we

prove 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 that

N(f) = k and f(z) 6= 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 satises jf(z)j > 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 2 f1; 2; 3; n3; n2; n1g,

for which no such f0; 1g{polynomial of degree n exists: such pairs

are related to regular (real and complex) Pisot numbers.

Similar, but less complete results for f1; 1g polynomials are

established. 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 about

the statistical distribution of N(f) in the set of Newman and Lit-

tlewood polynomials.

Newman 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], we

prove 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 that

N(f) = k and f(z) 6= 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 satises jf(z)j > 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 2 f1; 2; 3; n3; n2; n1g,

for which no such f0; 1g{polynomial of degree n exists: such pairs

are related to regular (real and complex) Pisot numbers.

Similar, but less complete results for f1; 1g polynomials are

established. 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 about

the statistical distribution of N(f) in the set of Newman and Lit-

tlewood polynomials.

Original language | English |
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Pages (from-to) | 831-870 |

Number of pages | 40 |

Journal | Mathematics of computation |

Volume | 90.2021 |

Issue number | March |

DOIs | |

Publication status | E-pub ahead of print - 27 Oct 2019 |