CYCLOTOMIC FACTORS OF BORWEIN POLYNOMIALS

2019 ◽  
Vol 100 (1) ◽  
pp. 41-47
Author(s):  
BISWAJIT KOLEY ◽  
SATYANARAYANA REDDY ARIKATLA
Keyword(s):  
Unit Circle ◽  
Prime Divisor ◽  
Monic Polynomial ◽  
Cyclotomic Polynomial ◽  
Newman Polynomials

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

Mahler's Measure of a Polynomial in Function of the Number of its Coefficients

1991 ◽  
Vol 34 (2) ◽  
pp. 186-195 ◽  
Author(s):  
Edward Dobrowolski
Keyword(s):  
Unit Circle ◽  
Lower Bounds ◽  
Monic Polynomial ◽  
Classical Theorem

AbstractMahler's measure of a monic polynomial is equal to the product of modules of its roots which lie outside the unit circle. By classical theorem of Kronecker it is strictly greater than 1 for any polynomial that is not a product of cyclotomic factors. In this case a number of lower bounds of the measure, depending either on the degree of the polynomial or on the number of its non-zero coefficients, has been found. Here is given an improvement of the bound of the latter type previously found by the author, A. Schinzel and W. Lawton.

Some Formulas Involving Ramanujan Sums

1962 ◽  
Vol 14 ◽  
pp. 284-286 ◽  
Author(s):  
C. A. Nicol
Keyword(s):  
Monic Polynomial ◽  
Cyclotomic Polynomial ◽  
Roots Of Unity ◽  
Ramanujan Sums ◽  
Image Position ◽  
Positive Integers

The purpose of this note is to establish an identity involving the cyclotomic polynomial and a function of the Ramanujan sums. Some consequences are then derived from this identity.For the reader desiring a background in cyclotomy, (2) is mentioned. Also, (4) is intimately connected with the following discussion and should be consulted.The cyclotomic polynomial Fn(x) is defined as the monic polynomial whose roots are the primitive nth roots of unity. It is well known that2.1For the proof of Corollary 3.2 it is mentioned that Fn(0) = 1 if n > 1 and that Fn(x) > 0 if |x| < 1 and 1 < n.The Ramanujan sums are defined by2.2where the sum is taken over all positive integers r less than or equal to n and relatively prime to n. It is also well known that2.3where the sum is taken over all positive divisors d common to n and k.

Each univariate complex polynomial has a ‘big’ factor

1994 ◽  
Vol 124 (1) ◽  
pp. 71-76 ◽  
Author(s):  
Ph. Glesser ◽  
M. Mignotte ◽  
M. Petkovic
Keyword(s):  
Recent Result ◽  
Unit Circle ◽  
General Result ◽  
Maximum Size ◽  
Monic Polynomial ◽  
Complex Polynomial ◽  
Complex Coefficients

We consider the following problem: Let P be a monic polynomial of degree n with complex coefficients. What can be the maximum ‘size’ of a monic divisor Q of P? Here the size of a polynomial R is the maximum ||R|| of the moduli of its values on the unit circle. In 1991, B. Beauzamy proved that there exists a divisor Q with ||Q|| ≧ e∈n−1, ∈ = 0.0019, when all the roots of P belong to the unit circle. Using a very recent result of D. Boyd, we obtain a general result which, in the same case, gives ||Q||≧βn; here β = 1.38135 … is optimal.

Newman Polynomials, Reducibility, and Roots on the Unit Circle

Integers ◽  
2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Idris Mercer
Keyword(s):  
Unit Circle ◽  
Newman Polynomials

Abstract.A    We also show that certain plausible conjectures imply that the proportion of length 5 Newman polynomials with roots on the unit circle is

The Restricted Arc-Width of a Graph

10.37236/1734 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
David Arthur
Keyword(s):  
Unit Circle ◽  
Single Point ◽  
Minimum Width ◽  
Graph Operations ◽  
Function Mapping

An arc-representation of a graph is a function mapping each vertex in the graph to an arc on the unit circle in such a way that adjacent vertices are mapped to intersecting arcs. The width of such a representation is the maximum number of arcs passing through a single point. The arc-width of a graph is defined to be the minimum width over all of its arc-representations. We extend the work of Barát and Hajnal on this subject and develop a generalization we call restricted arc-width. Our main results revolve around using this to bound arc-width from below and to examine the effect of several graph operations on arc-width. In particular, we completely describe the effect of disjoint unions and wedge sums while providing tight bounds on the effect of cones.

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

scholarly journals Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis

Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1488
Author(s):  
Damian Trofimowicz ◽  
Tomasz P. Stefański
Keyword(s):  
Unit Circle ◽  
Phase Analysis ◽  
Characteristic Equation ◽  
Digital Filter ◽  
Digital Filters ◽  
Computing Time ◽  
Complex Function ◽  
Stochastic Methods ◽  
Final Population ◽  
Candidate Regions

In this paper, novel methods for the evaluation of digital-filter stability are investigated. The methods are based on phase analysis of a complex function in the characteristic equation of a digital filter. It allows for evaluating stability when a characteristic equation is not based on a polynomial. The operation of these methods relies on sampling the unit circle on the complex plane and extracting the phase quadrant of a function value for each sample. By calculating function-phase quadrants, regions in the immediate vicinity of unstable roots (i.e., zeros), called candidate regions, are determined. In these regions, both real and imaginary parts of complex-function values change signs. Then, the candidate regions are explored. When the sizes of the candidate regions are reduced below an assumed accuracy, then filter instability is verified with the use of discrete Cauchy’s argument principle. Three different algorithms of the unit-circle sampling are benchmarked, i.e., global complex roots and poles finding (GRPF) algorithm, multimodal genetic algorithm with phase analysis (MGA-WPA), and multimodal particle swarm optimization with phase analysis (MPSO-WPA). The algorithms are compared in four benchmarks for integer- and fractional-order digital filters and systems. Each algorithm demonstrates slightly different properties. GRPF is very fast and efficient; however, it requires an initial number of nodes large enough to detect all the roots. MPSO-WPA prevents missing roots due to the usage of stochastic space exploration by subsequent swarms. MGA-WPA converges very effectively by generating a small number of individuals and by limiting the final population size. The conducted research leads to the conclusion that stochastic methods such as MGA-WPA and MPSO-WPA are more likely to detect system instability, especially when they are run multiple times. If the computing time is not vitally important for a user, MPSO-WPA is the right choice, because it significantly prevents missing roots.

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