A new approach to construct a Universal Groebner Basis and a Universal Parametric Groebner Basis

The talk will present a simple way to construct a universal Groebner basis of a polynomial ideal which serves as a Groebner basis for all admissible term orderings. A byproduct is the approach is a new way for basis conversion of a Groebner basis for one ordering to another ordering irrespective of dimensionality of the associated ideal. This construction of a universal Groebner basis is integrated with parametric (more popularly called comprehensive) Groebner basis which serves as a Groebner basis for all possible specializations of parameters. Two related approaches will be presented. First one extends Kapur's algorithm for computing a parametric Groebner basis in which along with branching based on making the head coefficient nonzero or not, branching on ordering constraints is also done in order first to choose a term that could serve as the head term. The second one is based on the Kapur, Sun and Wang's algorithm for computing comprehensive Groebner basis and system but uses a reduced universal Groebner basis to generate a universal parametric Groebner basis. The result of these algorithm is a mega Groebner basis that works for every admissible ordering as well as for any specialization of parameters.