Minimal polynomials for compact sets of the complex plane

LetW _N(z)=a_Nz^N+... be a complex polynomial and letT _n be the classical Chebyshev polynomial. In this article it is shown that the polynomials (2a_N)^?n+1T_n(W_N), n ∈N, are minimal polynomials on all equipotential lines for {z∈C:|W _N(z)|≤1 Λ ImW _N(z)=0}     

On Dirichlet Type Problems of Polynomial Dirac Equations with Boundary Conditions

Let ${{\bf D}_{\bf x} := \sum_{i=1}^n \frac{\partial}{\partial x_i} e_i}$ be the Euclidean Dirac operator in ${\mathbb{R}^n}$ and let P(X) = a _m X ^m + . . . + a ₁ X +  a ₀ be a polynomial with real coefficients. Differential equations of the form P(D _x )u(x) = 0 are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slightly less general form P(D _x )u(x) = f(x) (where the roots are exclusively real) with prescribed boundary conditions that avoid blow-ups inside the domain. We set up analytic representation formulas for the solutions in terms of hypercomplex integral operators and give exact formulas for the integral kernels in the particular cases dealing with spherical and concentric annular domains. The Maxwell and the Klein–Gordon equation are included as special subcases in this context.     

Zero distribution and division results for exponential polynomials

An exponential polynomial of order q is an entire function of the form

$$g(z) = {P_1}(z){e^{{Q_1}(z)}} + ...{P_k}(z){e^{{Q_k}(z)}},$$

where the coefficients P_j(z),Q_j(z) are polynomials in z such that

$$\max \{ deg({Q_j})\} = q.$$

It is known that the majority of the zeros of a given exponential polynomial are in domains surrounding finitely many critical rays. The shape of these domains is refined by showing that in many cases the domains can approach the critical rays asymptotically. Further, it is known that the zeros of an exponential polynomial are always of bounded multiplicity. A new sufficient condition for the majority of zeros to be simple is found. Finally, a division result for a quotient of two exponential polynomials is proved, generalizing a 1929 result by Ritt in the case q = 1 with constant coefficients. Ritt’s result is closely related to Shapiro’s conjecture that has remained open since 1958.

     

Distance regularity of Kerdock codes

A code is called distance regular, if for every two codewords x, y and integers i, j the number of codewords z such that d(x, z) = i and d(y, z) = j, with d the Hamming distance, does not depend on the choice of x, y and depends only on d(x, y) and i, j. Using some properties of the discrete Fourier transform we give a new combinatorial proof of the distance regularity of an arbitrary Kerdock code. We also calculate the parameters of the distance regularity of a Kerdock code.     

Shifted Appell Sequences in Clifford Analysis

The aim of this paper is to present a generalization of the Appell sequences within the framework of Clifford analysis called shifted Appell sequences. It consists of sequences {M _n (x)} _n ≥ 0 of monogenic polynomials satisfying the Appell condition (i.e. the hypercomplex derivative of each polynomial in the sequence equals, up to a multiplicative constant, its preceding term) such that the first term M ₀(x) = P _k (x) is a given but arbitrary monogenic polynomial of degree k defined in ${\mathbb{R}^{m+1}}$ . In particular, we construct an explicit sequence for the case ${M_0(x)=\mathbf{P}_k(\underline x)}$ being an arbitrary homogeneous monogenic polynomial defined in ${\mathbb R^m}$ . The connection of this sequence with the so-called Fueter’s theorem will also be discussed.     

Further Pieri-type formulas for the nonsymmetric Macdonald polynomial

The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial P _?? (z) are known explicitly. These formulas generalise the known r=1 case of the Pieri-type formulas for the nonsymmetric Macdonald polynomials E _?? (z). In this paper, we extend beyond the case r=1 for the nonsymmetric Macdonald polynomials, giving the full generalisation of the Pieri-type formulas for symmetric Macdonald polynomials. The decomposition also allows the evaluation of the generalised binomial coefficients $\tbinom{\eta }{\nu }_{q,t}$ associated with the nonsymmetric Macdonald polynomials.     

Palindromic and perturbed polynomials: zeros location

We give necessary and sufficient conditions for all zeros of the palindromic polynomial R(z)=1+λ(z+z ²+…+z ^n?1)+z ⁿ , \(\lambda\in \mathbb{R}\) , to be on the unit circle and we find \(\gamma \in \mathbb{R}\) for which S(z)=R(z)+γz ⁿ , n natural, has all its zeros in the closed unit disc.     

Bounds for Zeros of Solutions of Second Order Differential Equations with Polynomial Coefficients

We consider the equation ${u''=P(z)u\;\;(z\in\mathbb{C})}$ where P(z) is a polynomial. Let z _k (u), k = 1, 2,... be the zeros of a solution u(z) to that equation. Bounds for the sums $$\sum_{k=1}^{j} \frac {1} {|z_k(u)|}\;(j=1, 2, \ldots)$$ are established. Some applications of these bounds are also considered.     

A proof of a theorem by Fried and MacRae and applications to the composition of polynomial functions

Fried and MacRae (Math. Ann. 180, 220?C226 (1969)) proved that for univariate polynomials ${p,q, f, g \in \mathbb{K}[t]}$ ( ${\mathbb{K}}$ a field) with p, q nonconstant, p(x) ? q(y) divides f(x) ? g(y) in ${\mathbb{K}[x,y]}$ if and only if there is ${h \in \mathbb{K}[t]}$ such that f?=?h(p(t)) and g?=?h(q(t)). Schicho (Arch. Math. 65, 239?C243 (1995)) proved this theorem from the viewpoint of category theory, thereby providing several generalizations to multivariate polynomials. In the present note, we give a new proof of one of these generalizations. The theorem by Fried and MacRae yields a way to prove the following fact for nonconstant functions f, g from ${\mathbb{C}}$ to ${\mathbb{C}}$ : if both the composition ${f \circ g}$ and g are polynomial functions, then f has to be a polynomial function as well. We give an algebraic proof of this fact and present a generalization to multivariate polynomials over algebraically closed fields. This provides a way to prove a generalization of a result by Carlitz (Acta Sci. Math. (Szeged) 24, 196?C203 (1963)) that describes those univariate polynomials over finite fields that induce bijective functions on all of their finite extensions.     

Polynomials That Force a Unital Ring to be Commutative

We characterize polynomials f with integer coefficients such that a ring with unity R is necessarily commutative if f(R) = 0, in the sense that f(x) = 0 for all ${x \in R}$ . Such a polynomial must be primitive, and for primitive polynomials the condition f(R) = 0 forces R to have nonzero characteristic. The task is then reduced to considering rings of prime power characteristic and the main step towards the full characterization is a characterization of polynomials f such that R is necessarily commutative if f(R) = 0 and R is a unital ring of characteristic some power of a fixed prime p.     

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

On Some Zero-Increasing Operators

We characterize the set of linear operators of the form $$Tx^n = x^n + \sum\limits_{k = 0}^{n - 1} {b_{n,k} x^k ,} n = 0,1, \ldots ,$$ which satisfy Z(T _p ) ≧ Z(p) for every polynomial p, where Z counts the number of zeros on all of R. We also consider the analogous question on [0,∞)].     

Fundamental solution for linear two-point boundary value problem

The quadratic functional minimization with differential restrictions represented by the command linear systems is considered. The optimal solution determination implies the solving of a linear problem with two points boundary values. The proposed method implies the construction of a fundamental solution S(t)—a n×n matrix—and of a vector h(t) defining an adjoint variable λ(t) depending of the state variable x(t). From the extremum necessary conditions it is obtained the Riccati matrix differential equation having the S(t) as unknown fundamental solution is obtained. The paper analyzes the existence of the Riccati equation solution S(t) and establishes as the optimal solution of the proposed optimum problem. Also a superior limit of the minimum for the considered quadratic functionals class are evaluated.     

Time-Varying Fractionally Integrated Processes with Finite or Infinite Variance and Nonstationary Long Memory

Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x _t =∑ _j=0 ^∞ a _j (t)x _t?j , B(d)x _t =∑ _j=0 ^∞ b _j (t)x _t?j depending on arbitrary given sequence d=(d _t ,t∈?) of real numbers, such that A(d)^?1=B(?d), B(d)^?1=A(?d) and such that when d _t ≡d is a constant, A(d)=B(d)=(1?L) ^d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X _t =B(d)ε _t and Y _t =A(d)ε _t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d _±=lim? _t→±∞ d _t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε _t ,t∈? are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d _t ,t∈?) into a single class of sequences d=(d _t ,t∈?) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X _t and Y _t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments.     

A note on the reducibility of binary affine polynomials

Stickelberger–Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial. Based on this theorem, we prove in this note that every affine polynomial A(x) over ${\mathbb{F}_2}$ with degree >1, where A(x) = L(x) + 1 and ${L(x)=\sum_{i=0}^{n}{x^{2^i}}}$ is a linearized polynomial over ${\mathbb{F}_2}$ , is reducible except x ² + x + 1 and x ⁴ + x + 1. We also give some explicit factors of some special affine pentanomials over ${\mathbb{F}_2}$ .     

On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ?? n, given by F ₁(z)?=?(0, I _n;m (z)) with the isometric constant equal to (m?+?1)/(n?+?1) and F ₂(z)?=?(I _n;m (z), I _n;m (z)) with the isometric constant equal to 2(m?+?1)/(n?+?1). Here ${I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}$ is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.     

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.