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
A generalization of Gr(?)bner bases over ring (Z/(pe)[x1,…,xn])[I is given, where Z is the ring of integers, p is a prime, e≥1, and I is an ideal of Z/(pe)[x1,…,xn]. By applying this generalization, an algorithm is...A generalization of Gr(?)bner bases over ring (Z/(pe)[x1,…,xn])[I is given, where Z is the ring of integers, p is a prime, e≥1, and I is an ideal of Z/(pe)[x1,…,xn]. By applying this generalization, an algorithm is presented, which can synthesize multisequence with an equal or unequal length over Z[(m). The computational complexity of this algorithm is O(N2).展开更多
For a simple undirected graph G, denote by λ（G） the （0, 1）-adjacency matrix of G. Let the matrix S（G） = J-I-2A（G） be its Seidel matrix, and let SG（A） = det（AI-S（G）） be its Seidel characteristic polynomi...For a simple undirected graph G, denote by λ（G） the （0, 1）-adjacency matrix of G. Let the matrix S（G） = J-I-2A（G） be its Seidel matrix, and let SG（A） = det（AI-S（G）） be its Seidel characteristic polynomial, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. If all eigenvalues of SG（λ） are integral, then the graph G is called S-integral, In this paper, our main goal is to investigate the eigenvalues of SG（A） for the complete multipartite graphs G = Kn1,n2,...,n,. A necessary and sufficient condition for the complete tripartite graphs Km,n,t and the complete multipartite graphs Km,.... m,n,...,n to be S-integral is given, respectively.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Project supported by the State Key Laboratory of Information SecurityGraduate School of the Chinese Academy of Sciences
文摘A generalization of Gr(?)bner bases over ring (Z/(pe)[x1,…,xn])[I is given, where Z is the ring of integers, p is a prime, e≥1, and I is an ideal of Z/(pe)[x1,…,xn]. By applying this generalization, an algorithm is presented, which can synthesize multisequence with an equal or unequal length over Z[(m). The computational complexity of this algorithm is O(N2).
基金Supported by the National Natural Science Foundation of China (No.60863006)Program for New Century Excellent Talents in University (No.06-0912)
文摘For a simple undirected graph G, denote by λ（G） the （0, 1）-adjacency matrix of G. Let the matrix S（G） = J-I-2A（G） be its Seidel matrix, and let SG（A） = det（AI-S（G）） be its Seidel characteristic polynomial, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. If all eigenvalues of SG（λ） are integral, then the graph G is called S-integral, In this paper, our main goal is to investigate the eigenvalues of SG（A） for the complete multipartite graphs G = Kn1,n2,...,n,. A necessary and sufficient condition for the complete tripartite graphs Km,n,t and the complete multipartite graphs Km,.... m,n,...,n to be S-integral is given, respectively.
