Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.     

Approximation by Algebraic Polynomials on Rectangles

A direct theorem for approximation by algebraic polynomials in two variables with different degrees in each variable in L_p-metric (1 p ) on rectangles is proved, and the dependence of the constants on various parameters is studied.     

On the computation of all extensions of a -adic field of a given degree

Let be a -adic field. It is well-known that has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions of a given degree and discriminant.

     

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

     

Bases for Cooperations in K-Theory

Gaussian polynomials are used to define bases with good multiplicative properties for the algebra K _*(K) of cooperations in K-theory and for the invariants under conjugation.     

Representations of 2-Groups by Polynomials

It is known that any finite p-group can be represented by polynomials. However, how to represent p-groups and how to classify p-groups up to isomorphism are interesting and open questions. In this article, we investigate the 2-groups of order 8, and represent the dihedral group D_2n, the generalized quaternion group Q_2n, and the infinite dihedral group D.2000 Mathematics Subject Classification: 20C99, 20E99     

The connection between self‐associated two‐dimensional vector functionals and third degree forms

We deal with the 2‐orthogonal, 2‐symmetric self‐associated sequence (2‐orthogonal Tchebychev polynomials) and its cubic components. We prove that all the forms (linear functionals) arising are third degree forms. Therefore, an introduction to third degree forms is provided. We look for the connection between these components which are 2‐orthogonal with respect to the functional vector ^t(w₀{^μ},w₁ ^μ) and orthogonal sequences with respect to w₀ ^μ, μ=0,1,2. Associated forms w₀ ^μ)¹) and their inverse w₀ ^μ)^-1 are also studied through the symmetrized w₀}₀ ^μ, μ=0,1,2. Further, we give integral representations for some of these forms. This revised version was published online in June 2006 with corrections to the Cover Date.     

On type basic hypergeometric orthogonal polynomials

The five parameter family of Koornwinder's multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable -Racah polynomials and multivariable big and little -Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known -analogues of the Selberg integral are obtained.

     

Avoidable algebraic subsets of Euclidean space

Fix an integer and consider real -dimensional . A partition of avoids the polynomial , where each is an -tuple of variables, if there is no set of the partition which contains distinct such that . The polynomial is avoidable if some countable partition avoids it. The avoidable polynomials are studied here. The polynomial is an especially interesting example of an avoidable one. We find (1) a countable partition which avoids every avoidable polynomial over , and (2) a characterization of the avoidable polynomials. An important feature is that both the ``master' partition in (1) and the characterization in (2) depend on the cardinality of .

     

q-Analogue of the Watanabe Unitary Transform Associated to the q-Continuous Gegenbauer Polynomials

We associate a family of Hilbert spaces H _q ^2;(D) of analytic functions on the unit disk D=z :|z|<1 the q-continuous Gegenbauer polynomials C _n (x;q) on the interval]–1;1[ and give a q-analogue of the unitary integral transform that Watanabe constructed from the Hilbert space L ²(]–1;1[;(1–x ²)^– dx onto the weighted Hilbert space H ^2;(D).