Related papers: Polynomial-exponential decomposition from moments
Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible…
We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in…
We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and…
Considering a differential operator of third order that does not increase the degree of polynomials, we analyse some properties of elements of the dual space of 2-orthogonal polynomial eigenfunctions. In two classes of such generic…
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is…
Multivariate orthogonal polynomials in $D$ real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials,…
We relate the $m$-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when $m=2$. Each {\sf g}-vector cone…
We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank $2$ Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from…
A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this…
The asymptotic performance of a polar code under successive cancellation decoding is determined by the exponent of its polarizing matrix. We first prove that the partial distances of a polarizing matrix constructed from the Kronecker…
Let $H$ be a hyperexponential function in $n$ variables $x=(x_1,\dots,x_n)$ with coefficients in a field $\mathbb{K}$, $[\mathbb{K}:\mathbb{Q}] <\infty$, and $\omega$ a rational differential $1$-form. Assume that $H\omega$ is closed and $H$…
The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way.…
We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval,…
We present an effective method for computing parametric primary decomposition via comprehensive Gr\"obner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with…
We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by…
We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth…
We introduce a new family of strong linearizations of matrix polynomials---which we call "block Kronecker pencils"---and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any…
Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…
We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner…
The reduced Kronecker coefficients are particular instances of Kronecker coefficients that contain enough information to recover them. In this notes we compute the generating function of a family of reduced Kronecker coefficients. We also…