Rational Functions In Several Variables Research Articles

Twisted Weyl group multiple Dirichlet series over the rational function field

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables, with analytic continuation to Cr and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank r. Such series may be defined for any global field K, but in the case when K is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when K=Fq(T), and describe the denominators and support of the numerators.

On a radical extension of the field of rational functions in several variables

Let F be a field and let be the field of rational functions in n variables over F. Let and let m be a positive integer such that . Is it possible to express each as a rational function in and T over F? It is not difficult to prove that this can be done but it is another matter to show how this is done. We answer the above question affirmatively with a nonconstructive proof and a constructive proof.

Fusion procedure for the walled Brauer algebra

We establish two versions of the fusion procedure for the walled Brauer algebras. In each of them, a complete system of primitive pairwise orthogonal idempotents for the walled Brauer algebra is constructed by consecutive evaluations of a rational function in several variables on contents of standard walled tableaux.

Gauss congruences for rational functions in several variables

We investigate necessary as well as sufficient conditions under which the Laurent series coefficients fn associated to a multivariate rational function satisfy the Gauss congruences, that is, fmpr≡fmpr−1 (mod pr). For instance, we show that these congruences hold for certain determinants of logarithmic derivatives. As an application, we completely classify rational functions P/Q satisfying the Gauss congruences when Q is linear in each variable.

Fusion procedure for Coxeter groups of type 𝐵 and complex reflection groups 𝐺(𝑚,1,𝑛)

A complete system of primitive pairwise orthogonal idempotents for the Coxeter groups of type B B and, more generally, for the complex reflection groups G ( m , 1 , n ) G(m,1,n) is constructed by a sequence of evaluations of a rational function in several variables with values in the group ring. The evaluations correspond to the eigenvalues of the two arrays of Jucys–Murphy elements.

Positivity of rational functions and their diagonals

The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szegő as well as Askey and Gasper, who inspired more recent work. It is well known that the diagonal coefficients of rational functions are D-finite. This note is motivated by the observation that, for several of the rational functions whose positivity has received special attention, the diagonal terms in fact have arithmetic significance and arise from differential equations that have modular parametrization. In each of these cases, this allows us to conclude that the diagonal is positive.Further inspired by a result of Gillis, Reznick and Zeilberger, we investigate the relation between positivity of a rational function and the positivity of its diagonal.

Fusion Procedure for Cyclotomic Hecke Algebras

A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.

Idempotents for Birman-Murakami-Wenzl algebras and reflection equation

A complete system of pairwise orthogonal minimal idempotents for Birman–Murakami–Wenzl algebras is obtained by a consecutive evaluation of a rational function in several variables on sequences of quantum contents of up-down tableaux. A byproduct of the construction is a one-parameter family of fusion procedures for Hecke algebras. Classical limits to two different fusion procedures for Brauer algebras are described.

A New Fusion Procedure for the Brauer Algebra and Evaluation Homomorphisms

We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. In particular, this provides a new fusion procedure for the symmetric group involving an arbitrary parameter. The R-matrices are solutions of the Yang--Baxter equation associated with the classical Lie algebras g_N of types B, C and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra B(g_N) to U(g_N) and show that the fusion procedure provides an equivalence between natural tensor representations of B(g_N) with the corresponding evaluation modules.

Fusion procedure for the Brauer algebra

It is shown that all primitive idempotents for the Brauer algebra $\mathcal {B}_n(\omega )$ can be found by evaluating a rational function in several variables that has the form of a product of $R$-matrix type factors. This provides an analog of the fusion procedure for $\mathcal {B}_n(\omega )$.

Reducibility of rational functions in several variables

We prove an analogous of Stein theorem for rational functions in several variables: we bound the number of reducible fibers by a formula depending on the degree of the fraction.

On Some Properties of Polynomial Bases of Subspaces over the Field of Rational Functions in Several Variables

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a “free” polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.

Arrangement of hyperplanes. I: Rational functions and Jeffrey-Kirwan residue

Consider the space R Δ of rational functions of several variables with poles on a fixed arrangement Δ of hyperplanes. We obtain a decomposition of R Δ as a module over the ring of differential operators with constant coefficients. We generalize the notions of principal part and of residue to the space R Δ, and we describe their relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of the work by L. Jeffrey and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.

On the strong type multiplier norms of rational functions in several variables

On the strong type multiplier norms of rational functions in several variables

Low complexity k-dimensional Taylor forms

A low cost algorithm for the Taylor expansion of the rational function in several variables is presented. The algorithm is applied to compute the first two terms of a sample rational function in two variables.

Centered forms for functions in several variables

R. E. Moore [3] has introduced the centered form of a rational function f for obtaining good and easily computable approximations which include the exact range of f over an interval X. Moore's definition is implicit, while explicit formulas for the centered form are given in [8] and [9]. In [9], centered forms of higher order for functions of one variable are developed which lead to better estimations than the centered forms defined originally. In this paper, centered forms of higher order for rational functions in several variables are explicitly defined. They also give better approximations than the centered forms defined originally.

Centered Forms

R. E. Moore (Interval Analysis, Prentice-Hall, 1966) has introduced the centered form for evaluating ranges of rational functions of several variables approximately. While Moore’s definition is implicit, in this note explicit formulas for the centered form are given. Centered forms of second and higher order are established which lead to better estimations than the centered form (of first order) existing heretofore.

A property of the Taylor expansion of a class of rational functions in several variables

A property of the Taylor expansion of a class of rational functions in several variables

Weighted permutation problems and Laguerre polynomials

The connection between integrals of products of Laguerre polynomials, power series coefficients of certain rational functions of several variables, and certain numbers of weighted permutation problems is investigated. A combinatorial proof of our main result would be very desirable since this could lead the way to more general result, q-analogs, and perhaps even a q-analog of MacMahon's Master Theorem.

Non-cyclic division algebras

We study the abelian symmetric subgroup of the Brauer group of a field. We investigate the Galois groups of subfields of division algebras of twisted rational functions in several variables and of iterated twisted Laurent series. We settle a conjecture of Kuyk on these algebras and give a new proof of a theorem of Amitsur on non-cyclic division algebras.