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

2020 ◽  
Vol 90 (328) ◽  
pp. 831-870
Author(s):  
Kevin G. Hare ◽  
Jonas Jankauskas
Keyword(s):  
Unit Disk ◽  
Number Of Zeros ◽  
Littlewood Polynomials

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.

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.

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

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