Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbne...Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a GrSbnerbasis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to GrSbner basis theory are reexamined and/or further developed for the parametric case： （i） Definition of a comprehensive Groebner basis, （ii） test for a comprehensive GrSbner basis, （iii） parameterized rewriting, （iv） S-polynomials among parametric polynomials, （v） completion algorithm for directly computing a comprehensive Groebner basis from a given basis of a parametric ideal. Elegant properties of Groebner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique GrSbner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Groebner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.展开更多
Weispfenning in 1992 introduced the concepts of comprehensive Gr?bner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then,this research ?eld has attracted muc...Weispfenning in 1992 introduced the concepts of comprehensive Gr?bner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then,this research ?eld has attracted much attention over the past several decades, and many effcient algorithms have been proposed. Moreover, these algorithms have been applied to many different ?elds,such as parametric polynomial equations solving, geometric theorem proving and discovering, quanti?er elimination, and so on. This survey brings together the works published between 1992 and 2018, and we hope that this survey is valuable for this research area.展开更多
基金supported by the National Science Foundation under Grant No.DMS-1217054
文摘Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a GrSbnerbasis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to GrSbner basis theory are reexamined and/or further developed for the parametric case： （i） Definition of a comprehensive Groebner basis, （ii） test for a comprehensive GrSbner basis, （iii） parameterized rewriting, （iv） S-polynomials among parametric polynomials, （v） completion algorithm for directly computing a comprehensive Groebner basis from a given basis of a parametric ideal. Elegant properties of Groebner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique GrSbner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Groebner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.
基金supported in part by the CAS Project QYZDJ-SSW-SYS022the National Natural Science Foundation of China under Grant No.61877058the Strategy Cooperation Project AQ-1701
文摘Weispfenning in 1992 introduced the concepts of comprehensive Gr?bner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then,this research ?eld has attracted much attention over the past several decades, and many effcient algorithms have been proposed. Moreover, these algorithms have been applied to many different ?elds,such as parametric polynomial equations solving, geometric theorem proving and discovering, quanti?er elimination, and so on. This survey brings together the works published between 1992 and 2018, and we hope that this survey is valuable for this research area.
