Generalized Bezoutian and the inversion problem for block matrices,I. General scheme

The unified approach to the matrix inversion problem initiated in this work is based on the concept of the generalized Bezoutian for several matrix polynomials introduced earlier by the authors. The inverse X^–1 of a given block matrix X is shown to generate a set of matrix polynomials satisfying certain conditions and such that X^–1 coincides with the Bezoutian associated with that set. Thus the inversion of X is reduced to determining the underlying set of polynomials. This approach provides a fruitful tool for obtaining new results as well as an adequate interpretation of the known ones.     

Critical points and values of complex polynomials

This paper is primarily concerned with complex polynomials which have critical points which are also fixed points. We show that certain perturbations of a critical fixed point satisfy an inequality. This inequality permits us to prove a local version of Smale's mean value conjecture. We also use Thurston's topological characterization of critically finite rational mappings to enumerate explicitly as branched mappings the set of complex polynomials which have all their critical points fixed.     

On Value Sets of Polynomials over a Finite Field

We study value sets of polynomials over a finite field, and value sets associated to pairs of such polynomials. For example, we show that the value sets (counting multiplicities) of two polynomials of degree at mostdare identical or have at mostq−(q−1)/dvalues in common whereqis the number of elements in the finite field. This generalizes a theorem of D. Wan concerning the size of a single value set. We generalize our result to pairs of value sets obtained by restricting the domain to certain subsets of the field. These results are preceded by results concerning symmetric expressions (of low degree) of the value set of a polynomial. K. S. Williams, D. Wan, and others have considered such expressions in the context of symmetric polynomials, but we consider (multivariable) polynomials invariant under certain important subgroups of the full symmetry group.     

A note on resultants

An explicit technique is developed for the calculation of the number of common zeros of a set of polynomials. The number of common zeros is determined by the vanishing of certain resultantlike polynomials.     

On multiplicative independence of rational function iterates

We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao’s method for constructing elements in the finite field $${\mathbb {F}}_{q^n}$$ whose orders are larger than any polynomial in n when n becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.     

On the pointwise limits of bivariate lagrange projectors

A linear algebra proof is given of the fact that the nullspace of a finite-rank linear projector, on polynomials in two complex variables, is an ideal if and only if the projector is the bounded pointwise limit of Lagrange projectors, i.e., projectors whose nullspace is a radical ideal, i.e., the set of all polynomials that vanish on a certain given finite set. A characterization of such projectors is also given in the real case. More generally, a characterization is given of those finite-rank linear projectors, on polynomials in d complex variables, with nullspace an ideal that are the bounded pointwise limit of Lagrange projectors. The characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences.     

Orthogonal Polynomials Satisfying Higher-Order Difference Equations

We introduce a large class of measures with orthogonal polynomials satisfying higher-order difference equations with coefficients independent of the degree of the polynomials. These measures are constructed by multiplying the discrete classical weights of Charlier, Meixner, Krawtchouk, and Hahn by certain variants of the annihilator polynomial of a finite set of numbers.     

ON HERMITE MATRIX POLYNOMIALS AND HERMITE MATRIX FUNCTIONS

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.     

Tight Frames of Polynomials and the Truncated Trigonometric Moment Problem

A simple parametrization is given for the set of positive measures with finite support on the circle group T that are solutions of the truncated trigonometric moment problem: where the parameters are, up to nonzero multiplicative constants, the polynomials whose roots all have a modulus less than one. This result is then used to characterize, on a certain natural Hilbert space of polynomials associated with the problem, all finite "weighted" tight frames of evaluation polynomials. Finally, a new and simple way of parametrizing the whole set of positive Borel measures on T, solutions of the given moment problem is deduced, via a limiting argument.     

Exponential polynomial reproducing property of non-stationary symmetric subdivision schemes and normalized exponential B-splines

An important capability for a subdivision scheme is the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modeling. In this regards, this study first provides necessary and sufficient conditions for a non-stationary subdivision to have the reproducing property of exponential polynomials. Then, the approximation order of such non-stationary schemes is discussed to quantify their approximation power. Based on these results, we see that the exponential B-spline generates exponential polynomials in the associated spaces, but it may not reproduce any exponential polynomials. Thus, we present normalized exponential B-splines that reproduce certain sets of exponential polynomials. One interesting feature is that the set of exponential polynomials to be reproduced is varied depending on the normalization factor. This provides us with the necessary accuracy and flexibility in designing target curves and surfaces. Some numerical results are presented to support the advantages of the normalized scheme by comparing them to the results without normalization.     

Some results for a class of generalized polynomials

A class of generalized polynomials is considered consisting of the null spaces of certain differential operators with constant coefficients. This class strictly contains ordinary polynomials and appropriately scaled trigonometric polynomials. An analog of the classical Bernstein operator is introduced and it is shown that generalized Bernstein polynomials of a continuous function converge to this function. A convergence result is also proved for degree elevation of the generalized polynomials. Moreover, the geometric nature of these functions is discussed and a connection with certain rational parametric curves is established. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bounds on degrees of p-adic separating polynomials

We study a discrete optimization problem introduced by Babai, Frankl, Kutin, and Štefankovi? (2001), which provides bounds on degrees of polynomials with p-adically controlled behavior. Such polynomials are of particular interest because they furnish bounds on the size of set systems satisfying Frankl-Wilson-type conditions modulo prime powers, with lower degree polynomials providing better bounds. We elucidate the asymptotic structure of solutions to the optimization problem, and we also provide an improved method for finding solutions in certain circumstances.     

A Generalization of a theorem of chang

Szigeti, Tuza and Révész have developed a method in [6] to obtain polynomial identities for the n×n matrix ring over a commutative ring starting from directed Eulerian graphs. These polynomials are called Euler-ian. In the first part of this paper we show some polynomials that are in the T-ideal generated by a certain set of Eulerian polynomials, hence we get some identities of the n×n matrices. This result is a generalization of a theorem of Chang [l]. After that, using this theorem, we show that any Eulerian identity arising from a graph which lias d-fold multiple edges follows from the standard identity of degree d     

关于勒让德多项式与契贝谢夫多项式间的关系

主要研究勒让德多项式与契贝谢夫多项式之间的关系的性质,利用生成函数和函数级数展开的方法,得出了勒让德多项式与契贝谢夫多项式之间的一个重要关系,这对勒让德多项式与契贝谢夫多项式的研究有一定的推动作用．     

Some Spectral Properties of Infinite Band Matrices

For operators generated by a certain class of infinite band matrices we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order recurrence relations. This enables us to describe some asymptotic behaviour of the corresponding systems of vector orthogonal polynomials. Finally, we provide some new convergence results for matrix Padé approximants.     

Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set

We present two strategies for producing and describing some connected non-locally connected Julia sets of infinitely renormalizable quadratic polynomials. By using a more general strategy, we prove that all of these Julia sets fail to be arc-wise connected, and that their critical point is non-accessible. Using the first strategy we prove the existence of polynomials having an explicitly given external ray accumulating two particular, symmetric points. The limit Julia set resembles in a certain way the classical non-locally connected set: “the topologists spiral.”     

Quadrature rules based on partial fraction expansions

Quadrature rules are typically derived by requiring that all polynomials of a certain degree be integrated exactly. The nonstandard issue discussed here is the requirement that, in addition to the polynomials, the rule also integrates a set of prescribed rational functions exactly. Recurrence formulas for computing such quadrature rules are derived. In addition, Fejér's first rule, which is based on polynomial interpolation at Chebyshev nodes, is extended to integrate also rational functions with pre-assigned poles exactly. Numerical results, showing a favorable comparison with similar rules that have been proposed in the literature, are presented. An error analysis of a representative test problem is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

Enumerative properties of shifted-Dyck partitions

We study the enumerative properties of a new class of (skew) shifted partitions. This class arises in the computation of certain parabolic Kazhdan-Lusztig polynomials and is closely related to ballot sequences. As consequences of our results, we obtain new identities for the parabolic Kazhdan-Lusztig polynomials of Hermitian symmetric pairs and for the ordinary Kazhdan-Lusztig polynomials of certain Weyl groups.     

The vanishing ideal of a finite set of closed points in affine space

Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger-Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.