Complex Polynomial Research Articles

A new property of arbitrary complex polynomials

A new property of an arbitrary complex polynomial P ( z ) related to its derivative is established. It turns out that for arbitrary three values a 1 , a 2 and a 3 , there are lower bounds for | P ( z ) | when z belongs to the set of a 1 , a 2 , a 3 points of P ( z ) .

A real moment-HSOS hierarchy for complex polynomial optimization with real coefficients

A real moment-HSOS hierarchy for complex polynomial optimization with real coefficients

Morse numbers of complex polynomials

Morse numbers of complex polynomials

Using Exponential Complex Polynomials for Constructing Closed Curves with Given Properties

In this paper, we present a method to compute the coefficients of a complex exponential polynomial of real argument that, while being decomposed into real and imaginary parts by Euler’s formula, obtains required interpolating and differential properties at any given points of its real graph. Moreover, imaginary components in their nodes of interpolation and differentiation serve as additional control tools that shape the polynomial appearance. Although the impact of these components is not yet studied extensively, we can still use them to achieve useful properties, e. g. we can minimize the total height of the polynomial graph. From the geometry standpoint, having these properties implies that the parametric curves constructed with such polynomials can go through given points, have predetermined tangent vectors in those or other points, and retain enough variability to have additional useful properties, for instance, the total length of these curves, or their maximal curvature can also be minimized within limits.

Value distribution of a pair of meromorphic functions

The well-known Picard theorem shows only what happens on the Picard exceptional value of a meromorphic function f. In this paper, we mainly consider what happens on the common or different Picard exceptional values of two or three meromorphic functions with certain types. For instance, we will provide some new results and discussions for the generalized Picard exceptional values or small functions of a pair of meromorphic functions fng(k) and gmf(k), these two functions are the crossed variants of the complex differential polynomials in Hayman's conjecture, where n, m are positive integers and k≥0. We give more details on the case that fg′ and gf′ when f and g are exponential polynomials in particular.

Some Inequalities Concerning Maximum Modulus of Complex Polynomials with Restricted Zeros

In this paper, certain new results concerning the maximum modulus of polynomials with restricted zeros are obtained. These estimates strengthen some well-known inequalities for polynomial due to Rivlin, Govil, Lal and others.

Application of a metric for complex polynomials to bounded modification of planar Pythagorean-hodograph curves

By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.

On the complex polynomial inequalities concerning some operators

On the complex polynomial inequalities concerning some operators

Blaschke Products and Smale’s Conjecture on Complex Polynomials

Blaschke Products and Smale’s Conjecture on Complex Polynomials

THE LIE ALGEBRA su(3) REPRESENTATION WITH RESPECT TO ITS BASIS

The eight-dimensional Lie algebra of 3×3 anti-Hermitian matrices withits traces equal to zero is denoted by su(3) whose Lie group is denoted by SU(3). Theresearch aims to provide all representations of su(3) with respect to its basis which isrealized on the three complex variables homogeneous polynomials P1 of degree three. The first step is to construct representations of SU(3) on the space H and the second step is to find all derived representations of SU(3). The obtained results are eight explicit formulas of representations su(3) ↷ P1.

A characterization of differential operators in the ring of complex polynomials

The paper aims to provide a full characterization of all operators T:P(ℂ)→P(ℂ) acting on the space of all complex polynomials that satisfy the Leibniz rule T(f⋅g)=T(f)⋅g+f⋅T(g)for all f,g∈P(ℂ). We do not assume the linearity of T. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form.

On the best approximation of analytic in a disk functions in the weighted Bergman space <i>B</i><sub>2,μ</sub>

Abstract. We obtain sharp inequalities between the best approximations of analytic in the unit disk functions by algebraic complex polynomials and the moduli of continuity of higher-order derivatives in the Bergman weighted space B2,μ. Based on these inequalities, the exact values of some known n-widths of classes of analytic in the unit disk functions are calculated.

Power-closed ideals of polynomial and Laurent polynomial rings

We investigate the structure of power-closed ideals of the complex polynomial ring R=C[x1,…,xd] and the Laurent polynomial ring R±=C[x1,…,xd]±=S−1C[x1,…,xd], where S is the multiplicatively closed semigroup S=[x1,…,xd]. Here, an ideal I is power-closed if f(x1,…,xd)∈I implies f(x1i,…,xdi)∈I for each natural number i. Important examples of such ideals are provided by the ideals of relations in Minkowski rings of convex polytopes. We investigate related closure and interior operators on the set of ideals of R and R± and we give a complete description of principal power-closed ideals and of radicals of general power-closed ideals of R and R±.

A new algorithm for computing the nearest polynomial to multiple given polynomials via weighted ℓ2,q-norm minimization and its complex extension

A new algorithm is proposed in this paper for computing the nearest polynomial to multiple given polynomials with a given zero in the real case, where the distance between polynomials is defined by the weighted ℓ2,q norm (0<q≤2). First, the problem is formulated as a univariate constrained non-convex minimization problem, where prior information of the coefficients of the polynomial can be embedded by selecting proper weights. Then, an iteratively reweighted algorithm is designed to solve the obtained problem, and also the convergence and rate of convergence are uniformly demonstrated for all q in (0,2]. Since all the existing methods for computing the nearest polynomial to multiple given polynomials with a given zero are limited to the real case, we ingeniously extend the results to the complex case. Finally, two representative examples that separately compute the nearest real and complex polynomials are presented to show the effectiveness of the proposed algorithm.

Saddle point braids of braided fibrations and pseudo-fibrations

Let gt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_t$$\\end{document} be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of gt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_t$$\\end{document} are distinct for all t, they form a braid B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_1$$\\end{document} on n strands. Likewise, if the critical points of gt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_t$$\\end{document} are distinct for all t, they form a braid B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document} on n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document} strands. In this paper we study the relationship between B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_1$$\\end{document} and B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document}. Composing the polynomials gt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_t$$\\end{document} with the argument map defines a pseudo-fibration map on the complement of the closure of B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_1$$\\end{document} in C×S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}\ imes S^1$$\\end{document}, whose critical points lie on B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document}. We prove that for B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_1$$\\end{document} a T-homogeneous braid and B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document} the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} and L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} there is a mixed polynomial f:C2→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:{\\mathbb {C}}^2\\rightarrow {\\mathbb {C}}$$\\end{document} in complex variables u, v and the complex conjugate v¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{v}$$\\end{document} such that both f and the derivative fu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_u$$\\end{document} have a weakly isolated singularity at the origin with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} as the link of the singularity of f and L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} as a sublink of the link of the singularity of fu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_u$$\\end{document}.

Liquid Sloshing in Soil-Supported Multiple Cylindrical Tanks Equipped with Baffle under Horizontal Excitation

The dynamic behavior of liquid storage tanks is one of the research issues about fluid–structure interaction problems. The analysis errors of the dynamics of multiple adjacent tanks can exist if neglecting soil–tank interaction since tanks are typically supported on flexible soil. In the present paper, the dynamics of a group of baffled cylindrical storage tanks supported on a circular surface foundation and undergoing horizontal excitation are analytically examined. For upper multiple tank–liquid–baffle subsystems, accurate solutions to the velocity potential for liquid sloshing are acquired according to the subdomain partition technique. A theoretical model is utilized to portray the continuous sloshing of each tank. For the soil–foundation subsystem, a lumped-parameter model is used to characterize the impacts of soil on upper-tank structures using Chebyshev complex polynomials that present the fitting results of horizontal, rocking, and coupling impedance functions. Then, a model of the soil–foundation–tank–liquid–baffle system is constructed on the basis of the substructure approach. The present sloshing frequencies, sloshing height, and hydrodynamic shear as well as the moment under rigid/soft soil foundations are compared to the available exact results and the numerical results to prove the validity of the present model. The error of the maximum sloshing height between the present and the numerical solutions is within 5.27%; the solution efficiency of system dynamics from the present model is 40–50 times faster than that from the ADINA model. A detailed parameter analysis of the dynamic characteristics and earthquake responses of the coupling system is presented. The research novelty is that an equivalent analytical model is presented, and it allows for investigating the dynamics of soil-supported multiple cylindrical tanks with a baffle, providing acceptable accuracy and high calculation efficiency.

Bernstein inequalities for quaternionic polynomials in the setting of generalized polynomials

Recently, several authors proved a Bernstein type inequality for so called quaternionic unilateral polynomials. Although these polynomials are not generalized complex polynomials between normed spaces, it will be proved in this contribution that their restrictions to each complex plane satisfy a Bernstein inequality for the real Fréchet derivative. This result will imply Bernstein's inequality for quaternionic unilateral polynomials which in addition holds for all complex norms in which the unit closed ball contains the real unit quaternion. The second main contribution consists in providing Bernstein-type inequalities for so called quaternionic canonical generalized polynomials. These polynomials have a factorized form which, due to the lack of commutativity for quaternionic multiplication leads to a completely different type of investigation comparing to the other class of quaternionic unilateral polynomials. However, Bernstein's inequality is proved for some remarkable classes of such polynomials of degree less than or equal to 3 and for a class of polynomials of arbitrary degree having at most two distinct roots.

On the Coincidence Theorem

We are proving Coincidence theorem due to Walsh for the case when the total degree of a polynomial is less than the number of arguments. Also, the following result has been proven: if $$p(z)$$ is a complex polynomial of degree $$n$$, then closed disk D that contains at least $$n-1$$ of its zeros (counting multiplicity) contains at least $$\left[\frac{n-2k+1}{2} \right]$$ zeros of its $$k$$-th derivative, provided that the arithmetical mean of these zeros is also centre of D. We also prove a variation of the classical composition theorem due to Szegö.

Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form QC(p)=apn+mp+c, where n≥2. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals.

On Variance and Average Moduli of Zeros and Critical Points of Polynomials

This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial.