Polynomials with coefficients from a finite set

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Javad Baradaran ◽  
Mohsen Taghavi
Keyword(s):  
Unit Circle ◽  
Open Disk ◽  
Complex Polynomials ◽  
Number Of Zeros ◽  
Littlewood Polynomials ◽  
Finite Set

AbstractThis paper focuses on the problem concerning the location and the number of zeros of those polynomials when their coefficients are restricted with special conditions. The problem of the number of the zeros of reciprocal Littlewood polynomials on the unit circle $\mathbb{T}$ is discussed, the interest on bounds for the number of the zeros of reciprocal polynomials on the unit circle arose after 1950 when Erdös began introducing problems on zeros of various types of polynomials. Our main result is the problem of finding the number of zeros of complex polynomials in an open disk.

On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk

2015 ◽  
Vol 67 (3) ◽  
pp. 507-526 ◽  
Author(s):  
Peter Borwein ◽  
Stephen Choi ◽  
Ron Ferguson ◽  
Jonas Jankauskas
Keyword(s):  
Unit Circle ◽  
Prime Number ◽  
Negative Coefficient ◽  
Explicit Formulas ◽  
Limit Values ◽  
Number Of Zeros ◽  
Littlewood Polynomials ◽  
Sign Change ◽  
Complex Zeros ◽  
Littlewood Polynomial

AbstractWe investigate the numbers of complex zeros of Littlewood polynomials p(z) (polynomials with coefficients {−1, 1}) inside or on the unit circle |z| = 1, denoted by N(p) and U(p), respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for N(p), U(p) for polynomials p(z) of these types. We show that if n + 1 is a prime number, then for each integer k, 0 ≤ k ≤ n − 1, there exists a Littlewood polynomial p(z) of degree n with N(p) = k and U(p) = 0. Furthermore, we describe some cases where the ratios N(p)/n and U(p)/n have limits as n → ∞ and find the corresponding limit values.

Efficient parallel rooting of complex polynomials on the unit circle

1991 ◽  
Vol 39 (10) ◽  
pp. 2347-2351 ◽  
Author(s):  
W.S. McCormick ◽  
J.L. Lansford
Keyword(s):  
Unit Circle ◽  
Complex Polynomials

scholarly journals The spectra of compact operators in Hilbert spaces

1965 ◽  
Vol 7 (1) ◽  
pp. 34-38
Author(s):  
T. T. West
Keyword(s):  
Hilbert Space ◽  
Unit Circle ◽  
Real Axis ◽  
Complex Plane ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
Conformal Mappings ◽  
The Real ◽  
Finite Set

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

scholarly journals On the Lq norm of cyclotomic Littlewood polynomials on the unit circle

2011 ◽  
Vol 151 (2) ◽  
pp. 373-384 ◽  
Author(s):  
TAMÁS ERDÉLYI
Keyword(s):  
Unit Circle ◽  
Main Tool ◽  
Factorization Theorem ◽  
Cyclotomic Polynomials ◽  
Littlewood Polynomials ◽  
Odd Degree ◽  
Image Position

Let n be the collection of all (Littlewood) polynomials of degree n with coefficients in {−1, 1}. In this paper we prove that if (P2ν) is a sequence of cyclotomic polynomials P2ν ∈ 2ν, then for every q > 2 with some a = a(q) > 1/2 depending only on q, where The case q = 4 of the above result is due to P. Borwein, Choi and Ferguson. We also prove that if (P2ν) is a sequence of cyclotomic polynomials P2ν ∈ 2ν, then for every 0 < q < 2 with some 0 < b = b(q) < 1/2 depending only on q. Similar results are conjectured for Littlewood polynomials of odd degree. Our main tool here is the Borwein–Choi Factorization Theorem.

scholarly journals Unimodular Roots of Special Littlewood Polynomials

2006 ◽  
Vol 49 (3) ◽  
pp. 438-447 ◽  
Author(s):  
Idris David Mercer
Keyword(s):  
Unit Circle ◽  
Minimum Modulus ◽  
Littlewood Polynomials ◽  
Littlewood Polynomial

AbstractWe call α(z) = a0 + a1z + · · · + an–1zn–1 a Littlewood polynomial if aj = ±1 for all j. We call α(z) self-reciprocal if α(z) = zn–1α(1/z), and call α(z) skewsymmetric if n = 2m + 1 and am+j = (–1)jam–j for all j. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in ℂ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.

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