Ying Guang Shi(1995 & 1999) obtained some quadratures, which is based onthe zeros of the so-called s-orthogonal polynomials with respect to some JacobiB.Bojanov(1996) and our recent work, we give here a simple and...Ying Guang Shi(1995 & 1999) obtained some quadratures, which is based onthe zeros of the so-called s-orthogonal polynomials with respect to some JacobiB.Bojanov(1996) and our recent work, we give here a simple and unified approachto these questions of this type and obtain quadratures in terms of the divided differ-ences, which is based on an appropriate representation of the Hermite interpolatingpolynomial, of corresponding function at the zeros of the appropriate s-orthogonalpolynomial with multiplicities.展开更多
Gr?bner basis theory for parametric polynomial ideals is explored with the main objective of mimicking the Gr?bner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive Gr?bner bas...Gr?bner basis theory for parametric polynomial ideals is explored with the main objective of mimicking the Gr?bner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive Gr?bner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a Gr?bner basis 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 Gr?bner basis theory are reexamined and/or further developed for the parametric case:(i) Definition of a comprehensive Gr?bner basis,(ii) test for a comprehensive Gr?bner basis,(iii) parameterized rewriting,(iv) S-polynomials among parametric polynomials,(v) completion algorithm for directly computing a comprehensive Gr?bner basis from a given basis of a parametric ideal. Elegant properties of Gr?bner bases in the classical ideal theory, such as for a fixed admissible term ordering,a unique Gr?bner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Gr?bner 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
文摘Ying Guang Shi(1995 & 1999) obtained some quadratures, which is based onthe zeros of the so-called s-orthogonal polynomials with respect to some JacobiB.Bojanov(1996) and our recent work, we give here a simple and unified approachto these questions of this type and obtain quadratures in terms of the divided differ-ences, which is based on an appropriate representation of the Hermite interpolatingpolynomial, of corresponding function at the zeros of the appropriate s-orthogonalpolynomial with multiplicities.
基金supported by the National Science Foundation under Grant No.DMS-1217054
文摘Gr?bner basis theory for parametric polynomial ideals is explored with the main objective of mimicking the Gr?bner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive Gr?bner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a Gr?bner basis 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 Gr?bner basis theory are reexamined and/or further developed for the parametric case:(i) Definition of a comprehensive Gr?bner basis,(ii) test for a comprehensive Gr?bner basis,(iii) parameterized rewriting,(iv) S-polynomials among parametric polynomials,(v) completion algorithm for directly computing a comprehensive Gr?bner basis from a given basis of a parametric ideal. Elegant properties of Gr?bner bases in the classical ideal theory, such as for a fixed admissible term ordering,a unique Gr?bner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Gr?bner 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.
