Abstract

Parametric Groebner basis and systems were proposed in 1990's independently by Weispfenning and Kapur to study solutions of parametric polynomials for various specializations of parameters. Kapur's motivation for studying them arose from the application of geometry theorem proving and model based image analysis.  Recently there is interest in using these structures for developing heuristics that first consider equalities over the complex field in a formula expressed using ordering relation with an objective of developing an incomplete method for solving problems formulated in the theory of real closed field. It is hoped this incomplete approach can handle a larger class of problems in practice than the cylinderical algebraic decomposition method by Collins and his collaborators.  We will give an overview of algorithms for computing parametric Groebner basis and system developed in collaboration with Profs. Sun and Wang of the Academy of Mathematics and System Science of the Chinese Academy of Sciences. An existence proof of a canonical comprehensive Groebner basis associated a parametric ideal will be presented. However, an algorithm to compute this object is still elusive.  Some open problems in this topic will be discussed.