Polynomial bases for algebraic elements

New Bases for Polynomial-Based Spaces

Since it is well-known that the Vandermonde matrix is ill-conditioned, while the interpolation itself is not unstable in function space, this paper surveys the choices of other new bases. These bases are data-dependent and are categorized into discretely l2-orthonormal and continuously L2-orthonormal bases. The first one construct a unitary Gramian matrix in the space l2(X) while the late...

Gröbner Bases of Algebraic Invariants in Polynomial Differential Systems

We consider polynomial differential systems with coefficients in a field K of characteristic zero and the action of a given group G of linear transformations on these systems. The fundamental idea is based on the test of membership in an ideal. We use Gröbner bases to describe the algebra of the covariants in relation to this group of transformations while developing an algorithmic method to co...

Algebraic Solution of Systems of Polynomial Equations Using Groebner Bases

Algebraic adjoint of the polynomials-polynomial matrix multiplication

This paper deals with a result concerning the algebraic dual of the linear mapping defined by the multiplication of polynomial vectors by a given polynomial matrix over a commutative field

Canonical Bases for Algebraic Computations

We investigate, for quotients of the non-commutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for our factor algebras, of particular interest for one-sided ideals, and show a few applications, e.g. how to compute (one-sided) syzygy modules.