We give first results of our investigation of the connection between Gröbner bases computation and Gaussian elimination. We show that for every input set F of polynomials a matrix of shifts of those polynomials exists such that by triangularizing this matrix we obtain a Gröbner basis of F .

Recent investigations in optimization theory concerning the structure of positive polynomials with a sparsity pattern are interpreted in the more invariant language of (iterated) fibre products of real algebraic varieties. This opens the perspective of treating on a unifying basis the cases of positivity on unbounded supports, on non-semialgebraic supports, or of polynomials depending on counta...

For random matrix ensembles with non-gaussian matrix elements that may exhibit some correlations, it is shown that centered traces of polynomials in the matrix converge in distribution to a Gaussian process whose covariance matrix is diagonal in the basis of Chebyshev polynomials. The proof is combinatorial and adapts Wigner’s argument showing the convergence of the density of states to the sem...

Divide-and-conquer method for interpolation in Guruswami-Sudan (GS) list decoding algorithm is considered. It is shown that the Groebner basis (GB) of the ideal of bivariate interpolation polynomials (IP) can be obtained as a product of trivariate polynomials corresponding to disjoint subsets of interpolation points. The impact of monomial ordering on the interpolation complexity is studied.

In this paper we present an implementation for computing bivariate dimension polynomials of finitely generated modules over a Weyl algebra in Maple. We recall some basic results in order to explain the notion of dimension polynomials and to introduce methods for their computation based on Gröbner basis techniques. We explain input options for the mentioned implementations and provide several ex...

We present a method for computing the dimension of C 1 piecewise polynomials on a triangulated polygonal domain in the plane. Our results verify a conjecture of Strang in a large number of cases. For fourth{degree piecewise polynomials we deene, inductively, a nodal basis on quite general meshes.

Formulae expressing explicitly the coefficients of an expansion of double Jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultraspher...

The article presents some results concerning H-bases and theirs applications in multivariate interpolation. We derived the space of reduced polynomials with respect to a particular inner product. We made some connections with least interpolation and presented two application of the connection between spaces of reduced polynomials modulo a H-basis and spaces of ideal interpolation. 2000 Mathemat...

3.1.1. Existence and uniqueness. Our immediate goal is to establish the existence of a sequence of orthogonal polynomials. Although we could, in principle, determine the coefficients a j of pn in the natural basis by using the orthogonality conditions (3.1.2), it is computationally advantageous to express pn in terms of lower-order orthogonal polynomials. Let us denote Pn := span { 1, x, x, · ·...

We present the SP package which provides some functions on Schubert polynomials. These polynomials contain as a subfamily the Schur functions that are the fundamental symmetric functions. They are indexed by permutations and provide a convenient basis of the polynomial ring.