Details:
Title  Canonical comprehensive Gröbner bases  Author(s)  Volker Weispfenning  Type  Article in Journal  Abstract  Comprehensive Gröbner bases for parametric polynomial ideals were introduced, constructed, and studied by the author in 1992. Since then the construction has been implemented in the computer algebra systems ALDES/SAC2, MAS, REDUCE and MAPLE. A comprehensive Gröbner basis is a finite subset G of a parametric polynomial ideal I such that σ(G) constitutes a Gröbner basis of the ideal generated by σ(I) under all specializations σ of the parameters in arbitrary fields. This concept has found numerous applications. In contrast to reduced Gröbner bases, however, no concept of a canonical comprehensive Gröbner basis was known that depends only on the ideal and the term order. In this note we find such a concept under very general assumptions on the parameter ring. After proving the existence and essential uniqueness of canonical comprehensive Gröbner bases in a nonconstructive way, we provide a corresponding construction for the classical case, where the parameter ring is a multivariate polynomial ring. It proceeds via the construction of a canonical faithful Gröbner system. We also prove corresponding results for canonical comprehensive Gröbner bases relative to specializations in a specified class Σ of fields. Some simple examples illustrate the features of canonical comprehensive Gröbner bases. Besides their theoretical importance, canonical comprehensive Gröbner bases are also of potential interest for efficiency reasons as indicated by the research of Montes.  Keywords  Gröbner bases, Gröbner systems, comprehensive Gröbner bases, uniformity in parameters  Length  16  ISSN  07477171 
File 
 Language  English  Journal  Journal of Symbolic Computation  Volume  36  Number  34  Pages  669683  Publisher  Academic Press, Inc.  Address  Duluth, MN, USA  Year  2003  Translation 
No  Refereed 
Yes  Organization 
Universität Passau 
