Recent Progress in the Study of Polynomials with Constrained Coefficients

Erdélyi, Tamás

doi:10.1007/978-3-030-37904-9_2

Cited by 1 publication

(3 citation statements)

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“…Similar trigonometric expressions exist for self-reciprocal and negative self-reciprocal polynomials of even and odd degrees. Due to these relations, the unimodular zeros of self-reciprocal polynomials are of interest for harmonic analysts, especially in conjunction with topics of the positivity and L p -norm estimation, see, for instance, [48,70,72,73]. For this purpose, let us define the function U (f ) := #{j, 1 ≤ j ≤ n : |α j | = 1}, that counts the unimodular zeros of f (z) in Eq.…”

Section: Introductionmentioning

confidence: 99%

“…Erdélyi [47] quickly improved 1/2 in the exponent of Sahasrabudhe's bound to 1 by combining the strengths of different methods used in [14,44,97]. For a more in-depth account refer to [48].…”

Section: Introductionmentioning

confidence: 99%

“…In the classical control theory, polynomials with unimodular roots are called singular: all root-counting methods, such as Schur-Cohn rule, Jury rule or Bistritz rule, singular polynomials require special, out-of-flow treatment [7]. From the results of [14,44,47,48,97] that were discussed above, it is clear that presently there are no known simple method to control U (f ) in Newman and Littlewood polynomials with the same finesse as N (f ).…”

Section: Introductionmentioning

confidence: 99%

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

Hare¹,

Jankauskas²

2019

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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.

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Section: Introductionmentioning

confidence: 99%