Multiplication of a Schubert polynomial by a Schur polynomial

Schur polynomials are a special case of Schubert polynomials. In this paper, we give an algorithm to compute the product of a Schubert polynomial with a Schur polynomial on the basis of Schubert polynomials. This is a special case of the general problem of the multiplication of two Schubert polynomials, where the corresponding algorithm is still missing. The main tools for the given algorithm is a factorization property of a special class of Schubert polynomials and the transition formula for Schubert polynomials.     

Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations

We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szegö recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [−1, 1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle.     

The Bezoutian and the algebraic Riccati equation

The classical Bezoutian is a square matrix which counts the common zeros of two polynomials in the complex plane. The usual proofs of this property are based on connections between the Bezoutian and the Sylvester resultant matrix. These proofs do not make transparent the nature of the Bezoutian as a finite dimensional operator. This paper establishes that the Bezoutian is a solution of a suitable operator Riccati equation which makes evident the connections between the Bezoutian as an operator and the common zeros of polynomials. One application to the inversion of block Hankel (Toeplitz) matrices is given. Brief discussions of other Bezoutian operators are included. Apparently, even in the classical case the connection between the Bezoutian and the Riccati equation has not been studied previously.     

The Bezoutian and the algebraic Riccati equation

The classical Bezoutian is a square matrix which counts the common zeros of two polynomials in the complex plane. The usual proofs of this property are based on connections between the Bezoutian and the Sylvester resultant matrix. These proofs do not make transparent the nature of the Bezoutian as a finite dimensional operator. This paper establishes that the Bezoutian is a solution of a suitable operator Riccati equation which makes evident the connections between the Bezoutian as an operator and the common zeros of polynomials. One application to the inversion of block Hankel (Toeplitz) matrices is given. Brief discussions of other Bezoutian operators are included. Apparently, even in the classical case the connection between the Bezoutian and the Riccati equation has not been studied previously.     

Schur function identities, their t-analogs, and k-Schur irreducibility

We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act trivially on the k-Schur functions, thus leading to a concept of irreducibility for these functions.     

Fast conversion algorithms for orthogonal polynomials

We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexity for the converse operation.     

An introduction to the k-defect polynomials

The 0-defect polynomial of a graph is just the chromatic polynomial. This polynomial has been widely studied in the literature. Yet little is known about the properties of k-defect polynomials of graphs in general, when 0 < k ≤ |E(G)|. In this survey we give some properties of k-defect polynomials, in particular we highlight the properties of chromatic polynomials which also apply to k-defect polynomials. We discuss further research which can be done on the k-defect polynomials.     

A semiclassical perspective on multivariate orthogonal polynomials

Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi-orthogonality conditions. We obtain several characterizations for these polynomials including the analogues of the semiclassical Pearson differential equation, the structure relation and a differential-difference equation.     

Stable Grothendieck polynomials and <Emphasis Type="Italic">K</Emphasis>-theoretic factor sequences

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565–596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood–Richardson rule of Buch (Acta Math 189(1):37–78, 2002). The proof is based on a generalization of the Robinson–Schensted and Edelman–Greene insertion algorithms. Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schützenberger (C R Acad Sci Paris Ser I Math 294(13):447–450, 1982). In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665–687, 1999) for the cohomological quiver polynomials.     

On euclidean domains

We consider Euclidean domains and their groups of units. Let K(a,b) be the set of remainders in the division of a by b. If Card K(a,b) = 1 for any a and b from a Euclidean domain R, then R is known to be isomorphic to the ring of polynomials over some field, see [4], [5]. On the other hand, the condition Card K(a,b) = 2 for any a and b implies that R is isomorphic to the ring Z of integers, see [2]. We give characterization of Euclidean domains and their groups of units under some other conditions on K(a,b).     

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.     

On Koornwinder classical orthogonal polynomials in two variables

In 1975, Tom Koornwinder studied examples of two variable analogues of the Jacobi polynomials in two variables. Those orthogonal polynomials are eigenfunctions of two commuting and algebraically independent partial differential operators. Some of these examples are well known classical orthogonal polynomials in two variables, such as orthogonal polynomials on the unit ball, on the simplex or the tensor product of Jacobi polynomials in one variable, but the remaining cases are not considered classical by other authors. The definition of classical orthogonal polynomials considered in this work provides a different perspective on the subject. We analyze in detail Koornwinder polynomials and using the Koornwinder tools, new examples of orthogonal polynomials in two variables are given.     

Brownian motion, quantum corrections and a generalization of the Hermite polynomials

The nonequilibrium evolution of a Brownian particle, in the presence of a “heat bath” at thermal equilibrium (without imposing any friction mechanism from the outset), is considered. Using a suitable family of orthogonal polynomials, moments of the nonequilibrium probability distribution for the Brownian particle are introduced, which fulfill a recurrence relation. We review the case of classical Brownian motion, in which the orthogonal polynomials are the Hermite ones and the recurrence relation is a three-term one. After having performed a long-time approximation in the recurrence relation, the approximate nonequilibrium theory yields irreversible evolution of the Brownian particle towards thermal equilibrium with the “heat bath”. For quantum Brownian motion, which is the main subject of the present work, we restrict ourselves to include the first quantum correction: this leads us to introduce a new family of orthogonal polynomials which generalize the Hermite ones. Some general properties of the new family are established. The recurrence relation for the new moments of the nonequilibrium distribution, including the first quantum correction, turns out to be also a three-term one, which justifies the new family of polynomials. A long-time approximation on the new three-term recurrence relation describes irreversible evolution towards equilibrium for the new moment of lowest order. The standard Smoluchowski equations for the lowest order moments are recovered consistently, both classically and quantum-mechanically.     

Lyapunov,Bézout,and Hankel

Given a polynomial f of degree n, we denote by C its companion matrix, and by S the truncated shift operator of order n. We consider Lyapunov-type equations of the form X?SXC=>W and X?CXS=W. We derive some properties of these equations which make it possible to characterize Bezoutian matrices as solutions of the first equation with suitable right-hand sides W (similarly for Hankel and the second equation) and to write down explicit expressions for these solutions. This yields explicit factorization formulae for polynomials in C, for the Schur-Cohn matrix, and for matrices satisfying certain intertwining relations, as well as for Bezoutian matrices.     

Bezoutian矩阵的一致逼近形式

借助闭区间上的连续函数可以用Bernstein 多项式一致逼近这一事实,将多项式对所生成的经典Bezoutian 矩阵和Bernstein Bezoutian 矩阵推广到C [0,1]上函数对所对应的情形,给出了 Bezoutian 矩阵一致逼近形式的定义,并且得到如下结论：给出了经典 Bezoutian 矩阵的 Barnett 型分解公式和三角分解公式的一致逼近形式;提供了经典Bezoutian 矩阵和Bernstein Bezoutian 矩阵的一致逼近形式的两类算法;得到了上述两种矩阵的一致逼近形式中元素间的两个恒等关系式。最后,利用数值实例对恒等关系式进行验证,结果表明两类算法是有效的。     

Bezoutian and Schur-Cohn problem for operator polynomials

A generalized Bezout operator (Bezoutian) for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials. The location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle (infinite-dimensional version of the Schur-Cohn problem) is expressed via the inertia of a suitable Bezoutian. An application to the geometric dichotomy problem for difference equations with operator coefficients is given as well.     

On some properties of symplectic Grothendieck polynomials

Grothendieck polynomials, introduced by Lascoux and Schützenberger, are certain K-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by Wyser and Yong, represent the K-theory classes of orbit closures for the complex symplectic group acting on the complete flag variety. We prove a transition formula for symplectic Grothendieck polynomials and study their stable limits. We show that each of the K-theoretic Schur P-functions of Ikeda and Naruse arises from a limiting procedure applied to symplectic Grothendieck polynomials representing certain “Grassmannian” orbit closures.     

The Bezoutian, state space realizations and Fisher’s information matrix of an ARMA process

In this paper we derive some properties of the Bezout matrix and relate the Fisher information matrix for a stationary ARMA process to the Bezoutian. Some properties are explained via realizations in state space form of the derivatives of the white noise process with respect to the parameters. A factorization of the Fisher information matrix as a product in factors which involve the Bezout matrix of the associated AR and MA polynomials is derived. From this factorization we can characterize singularity of the Fisher information matrix.     

A look-ahead algorithm for the solution of general Hankel systems

Summary The solution of systems of linear equations with Hankel coefficient matrices can be computed with onlyO(n ²) arithmetic operations, as compared toO(n ³) operations for the general cases. However, the classical Hankel solvers require the nonsingularity of all leading principal submatrices of the Hankel matrix. The known extensions of these algorithms to general Hankel systems can handle only exactly singular submatrices, but not ill-conditioned ones, and hence they are numerically unstable. In this paper, a stable procedure for solving general nonsingular Hankel systems is presented, using a look-ahead technique to skip over singular or ill-conditioned submatrices. The proposed approach is based on a look-ahead variant of the nonsymmetric Lanczos process that was recently developed by Freund, Gutknecht, and Nachtigal. We first derive a somewhat more general formulation of this look-ahead Lanczos algorithm in terms of formally orthogonal polynomials, which then yields the look-ahead Hankel solver as a special case. We prove some general properties of the resulting look-ahead algorithm for formally orthogonal polynomials. These results are then utilized in the implementation of the Hankel solver. We report some numerical experiments for Hankel systems with ill-conditioned submatrices.The research of the first author was supported by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).The research of the second author was supported in part by NSF grant DRC-8412314 and Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).     

Exceptional units and Euclidean number fields

By a result of H.W. Lenstra, one can prove that a number field is Euclidean with the aid of exceptional units. We describe two methods computing exceptional sequences, i.e., sets of units such that the difference of any two of them is still a unit. The second method is based on a graph theory algorithm for the maximum clique problem. This yielded 42 new Euclidean number fields in degrees 8, 9, 10, 11 and 12. Received: 16 May 2006