A note on biorthogonal ensembles

We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. These systems are closely related to multiple orthogonal functions. It is known that the eigenvalue correlation functions of such ensembles can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term.Comment: 20 page

Painleve IV and degenerate Gaussian Unitary Ensembles

We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.Comment: 17 page

Generalized companion matrix for approximate GCD

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]=(f). In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD(f; g). Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.Comment: Submitted to MEGA 201

A perturbed differential resultant based implicitization algorithm for linear DPPEs

Let \bbK be an ordinary differential field with derivation $\partial$. Let \cP be a system of $n$ linear differential polynomial parametric equations in $n-1$ differential parameters with implicit ideal \id. Given a nonzero linear differential polynomial $A$ in \id we give necessary and sufficient conditions on $A$ for \cP to be $n-1$ dimensional. We prove the existence of a linear perturbation \cP_{\phi} of \cP so that the linear complete differential resultant \dcres_{\phi} associated to \cP_{\phi} is nonzero. A nonzero linear differential polynomial in \id is obtained from the lowest degree term of \dcres_{\phi} and used to provide an implicitization algorithm for \cP