We consider matrices over a ring K [∂; σ, θ ] of Ore polynomials over a skew field K . Since the Popov and Hermite normal forms are both Gröbner bases (for term over position and position over term ordering resp.), the classical FGLM-algorithm provides a method of converting one into the other. In this report we investigate the exact formulation of the FGLM algorithm for not necessarily “zero-d...

We present an algorithm to compute the primary decomposition of any ideal in a polynomial ring over a factorially closed algorithmic principal ideal domain R. This means that the ring R is a constructive PID and that we are given an algorithm to factor polynomials over fields which are finitely generated over R or residue fields of R. We show how basic ideal theoretic operations can be performe...

The discrete Laguerre transform (DLT) belongs to the family of unitary transforms known as Gauss-Jacobi transforms. Using classical methodology, the DLT is derived from the orthonormal set of Laguerre functions. By examining the basis vectors of the transform matrix, the types of signals that can be best represented by the DLT are determined. Simulation results are used to compare the DLT's e e...

When a function is singular at the ends of its expansion interval, its Chebyshev coefficients a, converge very poorly. We analyze three numerical strategies for coping with such singularities of the form (1 + x)~ log(1 f x), and in the process make some modest additions to the theory of Chebyshev expansions. The first two numerical methods are the convergence-improving changes of coordinate x =...

Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive Fiedler-comrade pencils for two bases that are extremely important in practical ap...

We give a representation of the classical theory of multiplicative arithmetic functions (MF)in the ring of symmetric polynomials. The basis of the ring of symmetric polynomials that we use is the isobaric basis, a basis especially sensitive to the combinatorics of partitions of the integers. The representing elements are recursive sequences of Schur polynomials evaluated at subrings of the comp...

The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function Wγ(x) = x γ1 1 · · ·x γd d (1− |x|)d+1 when all γi > −1 and they are eigenfunctions of a second order partial differential operator Lγ . The singular cases that some, or all, γ1, . . . , γd+1 are −1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of Lγ ...

The main question of this paper is: When does Groebner basis computation (Buchberger, 1965, 1985) commute with composition? More precisely, let F be a finite set of polynomials in the variables x1, . . . , xn, and let G be a Groebner basis of the ideal generated by F under some term ordering. Let Θ = (θ1, . . . , θn) be a list of n polynomials in the variables x1, . . . , xn. Let F ∗ be the set...

We develop a method for approximating the Gröbner basis of the ideal of polynomials which vanish at a finite set of points, when the coordinates of the points are known with only limited precision. The method consists of a preprocessing phase of the input points to mitigate the effects of the input data uncertainty, and of a new “numerical” version of the Buchberger-Möller algorithm to compute ...