Complex Polynomials Of Degree Research Articles

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.

Generic classification and asymptotic enumeration of dope matrices

AbstractFor a complex polynomial of degree and an ‐tuple of distinct complex numbers , the dope matrix is defined as the matrix with if and otherwise. We classify the set of dope matrices when the entries of are algebraically independent, resolving a conjecture of Alon, Kravitz, and O'Bryant. We also provide asymptotic upper and lower bounds on the total number of dope matrices. For much smaller than , these bounds give an asymptotic estimate of the logarithm of the number of dope matrices.

Some Results Concerning Sendov Conjecture

Let P(z) be a complex polynomial of degree n having all its zeros in |z| ≤ 1. Then the Sendov’s Conjecture states that there is always a critical point of P(z) in |z − a| ≤ 1, where a is any zero of P(z). In this paper, we verify the Sendov’s Conjecture for some special cases. The case where a is the root of pth smallest modulus is also investigated.

Sharing a Measure of Maximal Entropy in Polynomial Semigroups

Abstract Let $P_1,P_2,\dots , P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or to Chebyshev polynomials, and $S$ the semigroup under composition generated by $P_1,P_2,\dots , P_k$. We show that all elements of $S$ share a measure of maximal entropy if and only if the intersection of principal left ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.

Continuity of the Roots of a Polynomial

We provide a self-contained topological proof of the continuous dependence of the roots of a polynomial on its coefficients. We present it as a homeomorphism between the space of monic complex polynomials of degree n and the space of unordered n-tuples of complex numbers. Our proof uses bounds on the roots of polynomials to show that the map that associates to each monic polynomial its roots has an inverse that is a proper map.

Spaces of Polynomials Related to Multiplier Maps

Let f(x) be a complex polynomial of degree n. We associate f with a ℂ-vector space W(f) that consists of complex polynomials p(x) of degree at most n — 2 such that f(x) divides f”(x)p(x) — f’(x)p’(x). The space W(f) first appeared in Yu. G. Zarhin’s work, where a problem concerning dynamics in one complex variable posed by Yu. S. Ilyashenko was solved. In this paper, we show that W(f) is nonvanishing if and only if q(x)2 divides f(x) for some quadratic polynomial q(x). In that case, W(f) has dimension (n — 1) — (n1 + n2 + 2N3) under certain conditions, where ni is the number of distinct roots of f with multiplicity i and N3 is the number of distinct roots of f with multiplicity at least 3.

Dynamics and limiting behavior of Julia sets of König's method for multiple roots

A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27]) shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f, there exists at least one point in Sd converging under iterations of Nf to α. Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton's method. We show that for all order σ≥2, there exists a complex polynomial f such that the Julia set of König's method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result in [27] to higher order root-finding methods. As far as we know, there are no pictures of disconnected Julia sets for root finding algorithms applied to polynomials. Here we give a proof and provide pictures that illustrate such disconnectedness. We also show that the Fatou set of König's method for multiple roots converges to the Voronoi diagram under order of convergence growth, in the Hausdorff complementary metric.

SPECTRAL PROPERTIES OF VOLTERRA-TYPE INTEGRAL OPERATORS ON FOCK-SOBOLEV SPACES

We study some spectral properties of Volterra-type integral operators $V_g$ and $I_g$ with holomorphic symbol $g$ on the Fock--Sobolev spaces $\mathcal{F}_{\psi_m}^p$. We showed that $V_g$ is bounded on $\mathcal{F}_{\psi_m}^p$ if and only if $g$ is a complex polynomial of degree not exceeding two, while compactness of $V_g$ is described by degree of $g$ being not bigger than one. We also identified all those positive numbers $p$ for which the operator $V_g$ belongs to the Schatten $\mathcal{S}_p$ classes. Finally, we characterize the spectrum of $V_g$ in terms of a closed disk of radius twice the coefficient of the highest degree term in a polynomial expansion of $g$.

A new family of spectrally arbitrary ray patterns

An n × n ray pattern A is called a spectrally arbitrary ray pattern if the complex matrices in Q(A) give rise to all possible complex polynomials of degree n.

On an inequality involving the complex polynomial and its derivative

Let P(z) be a complex polynomial of degree n of lacunary type having zeros in |z| < 1, and 1 ≤ r < R. In this paper, we shall estimate a new type of bound in more generalized form for \(\mathop {\max }\limits_{|z| = 1} |P'(z)|\) in terms of \(\mathop {\max }\limits_{|z| = Rr} |P(z)|\) and \(\mathop {\max }\limits_{|z| = R^2 } |P(z)|\).

A Simple Direct Proof of Marden's Theorem

Marden's theorem characterizes the critical points of complex polynomials of degree 3 in a nice geometrical way. Our proof of the theorem is based directly on the defining property of ellipses.

Simultaneous approximation by complex polynomials

Simultaneous approximation by complex polynomials

Geometry of the locus of polynomials of degree 4 with iterative roots

We study polynomial iterative roots of polynomials and describe the locus of complex polynomials of degree 4 admitting a polynomial iterative square root.

On the coefficients of a polynomial with restricted zeros

Let $$ P(z) = a_n z^n + a_n - 1z^{n - 1} + \cdots + a_0 $$ be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coefficients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z).

Bohr’s Radius for Polynomials in One Complex Variable

We consider complex polynomials of degree n that are bounded by one in the unit disc and give estimates on the size of the radius Rn of the disc where the sum of the moduli of the individual terms of the polynomial is less than one. We find that there are positive constants C1, C2 such that $$C_1 \frac{1} {{3^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} }} < R_n - \frac{1} {3} < C_2 \frac{{\log n}} {n}.$$ This result generalizes the celebrated theorem of Harald Bohr to polynomials of degree n.

Riemann–Hilbert analysis and uniform convergence of rational interpolants to the exponential function

We study the asymptotic behavior of the polynomials p and q of degrees n, rational interpolants to the exponential function, defined by p ( z ) e - z / 2 + q ( z ) e z / 2 = O ( ω 2 n + 1 ( z ) ) , as z tends to the roots of ω 2 n + 1 , a complex polynomial of degree 2 n + 1 . The roots of ω 2 n + 1 may grow to infinity with n, but their modulus should remain uniformly bounded by c log ( n ) , c < 1 / 2 , as n → ∞ . We follow an approach similar to the one in a recent work with Arno Kuijlaars and Walter Van Assche on Hermite–Padé approximants to e z . The polynomials p and q are characterized by a Riemann–Hilbert problem for a 2 × 2 matrix valued function. The Deift–Zhou steepest descent method for Riemann–Hilbert problems is used to obtain strong uniform asymptotics for the scaled polynomials p ( 2 nz ) and q ( 2 nz ) in every domain in the complex plane. From these asymptotics, we deduce uniform convergence of general rational interpolants to the exponential function and a precise estimate on the error function. This extends previous results on rational interpolants to the exponential function known so far for real interpolation points and some cases of complex conjugate interpolation points.

A Julia Set That Is Everything

We see unpredictable behavior around us every day: in the way weather patterns change, stock markets fluctuate, or wildfire spreads. We can also observe chaos in seemingly simple mathematical functions, and many recent undergraduate dynamical systems textbooks [5, 10, 11, 15] discuss chaotic systems. These texts all include chapters covering Julia sets, the part of the domain where a complex function behaves chaotically. A Julia set is usually an intricate and beautiful object, and images of Julia sets can be found on posters, book covers, T-shirts, screen savers, and web pages. Many books [10, 11, 15] focus on Julia sets of complex polynomials of degree 2 or 3, and even the casual reader will observe from the pictures that all of these Julia sets appear to have area zero. For polynomials, it turns out that there must always be some large region where the function has very predictable behavior. What if we examine other types of functions? Is there a complex function whose Julia set is everything? That is, does there exist a complex function that acts chaotically on its entire domain? The picture of such an example would be entirely black.

Differential properties for a class of Sobolev orthogonal polynomials

We study the orthogonal polynomials with respect to a Sobolev inner product of the following type: 〈f,g〉 s= ∫ 0 2π f( e iθ ) g( e iθ ) |B h( e iθ )| 2 dθ 2π + 1 λ ∫ 0 2π f′( e iθ ) g′( e iθ ) dθ 2π , z= e iθ , where B h ( z) is a complex polynomial of degree h, d θ/2 π is the normalized Lebesgue measure and λ is a positive real number. The asymptotic behavior in the complex plane, as well as the differential equations satisfied by the orthogonal polynomials are obtained. As an application, two differential problems are solved, one of them is like a Dirichlet boundary value problem.

On the Distribution of the Derivative Zeros of a Complex Polynomial

We consider the class S(n) of all complex polynomials of degree n > 1 having all their zeros in the closed unit disk Ē. By S(n,β) we denote the subclass of p ∈ S(n) vanishing in the prescribed point β ∈ Ē. For an arbitrary point α ∈ C and p ∈ S(n,β) let |p| α be the distance of α and the set of zeros of P'. Then there exists some P ∈ S(n,β) with maximal |P|α. We give an estimation for the number of zeros of P on |z| = 1$ resp. P' on $ |z-α| = |P| α .

Polynomials on the Cauchy circle

Let \(P\) be a complex polynomial of degree \(n\) with \(P(0)=1\) and Cauchy radius 1 about the origin. We discuss the order of magnitude of the minimal number \(N=N(\epsilon _n,n)\) such that\( \min_{1\leq k\leq N}|P({\rm e}^{2\pi {\rm i}k/N})|\leq 1-\epsilon _n. \) Previous estimates of \(N=O(n^{3/2})\) are improved to \(N= O(n\,\log\,n)\). Some other related properties of these polynomials are also exhibited.