The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only...The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only if f1, f2, …, fm are pairwise weakly relatively prime with λ1, λ2, …, λm arbitrary non-negative integers; polynomial composition by θ = （θ1,θ2, …, θn） commutes with monomial-Grobner bases computation if and only if θ1, θ2, , θm are pairwise weakly relatively prime.展开更多
Polynomial composition is the operation of replacing variables in a polynomial with other polynomials. λ-Grgbner basis is an especial Grobner basis. The main problem in the paper is： when does composition commute wi...Polynomial composition is the operation of replacing variables in a polynomial with other polynomials. λ-Grgbner basis is an especial Grobner basis. The main problem in the paper is： when does composition commute with λ-Grobner basis computation？ We shall answer better the above question. This has a natural application in the computation of λ-Grobner bases.展开更多
In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring fun...In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring function are involved, to have polynomial solutions. In order to judge whether the coupled functional equations are solvable, in this paper we give an algorithm to compute a Gr¨obner basis for irreducible decomposition of algebraic varieties so as to find algebraic relations among coefficients of the damping function and the restoring function. We demonstrate the algorithm for polynomial Li′enard systems of degree 5, which are divided into 25 cases. We find all conditions of those coefficients for the polynomial Li′enard center in 13 cases and prove that the origin is not a center in the other 12 cases.展开更多
It is a fundamental problem to determine the equivalence of indexed differential polynomials in both computer algebra and differential geometry.However,in the literature,there are no general computational theories for...It is a fundamental problem to determine the equivalence of indexed differential polynomials in both computer algebra and differential geometry.However,in the literature,there are no general computational theories for this problem.The main reasons are that the ideal generated by the basic syzygies cannot be finitely generated,and it involves eliminations of dummy indices and functions.This paper solves the problem by extending Grobner basis theory.The authors first present a division of the set of elementary indexed differential monomials E■ into disjoint subsets,by defining an equivalence relation on E■ based on Leibniz expansions of monomials.The equivalence relation on E■also induces a division of a Grobner basis of basic syzygies into disjoint subsets.Furthermore,the authors prove that the dummy index numbers of the sim-monomials of the elements in each equivalence class of E■ have upper bounds,and use the upper bounds to construct fundamental restricted rings.Finally,the canonical form of an indexed differential polynomial proves to be the normal form with respect to a subset of the Grobner basis in the fundamental restricted ring.展开更多
By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schube...By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk＇s formula.展开更多
基金Supported by the NSFC (10771058, 11071062, 10871205), NSFH (10JJ3065)Scientific Research Fund of Hunan Provincial Education Department (10A033)Hunan Provincial Degree and Education of Graduate Student Foundation (JG2009A017)
文摘The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only if f1, f2, …, fm are pairwise weakly relatively prime with λ1, λ2, …, λm arbitrary non-negative integers; polynomial composition by θ = （θ1,θ2, …, θn） commutes with monomial-Grobner bases computation if and only if θ1, θ2, , θm are pairwise weakly relatively prime.
基金The research is supported by the National Natural Science Foundation of China under Grant No. 10771058, Hunan Provincial Natural Science Foundation of China under Grant No. o6jj20053, and Scientific Research Fund of Hunan Provincial Education Department under Grant No. 06A017.
文摘Polynomial composition is the operation of replacing variables in a polynomial with other polynomials. λ-Grgbner basis is an especial Grobner basis. The main problem in the paper is： when does composition commute with λ-Grobner basis computation？ We shall answer better the above question. This has a natural application in the computation of λ-Grobner bases.
基金National Natural Science Foundation of China(Grant No.11371264)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20120181110062)Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme(Grant No.FP7-PEOPLE-2012-IRSES-316338)
文摘In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring function are involved, to have polynomial solutions. In order to judge whether the coupled functional equations are solvable, in this paper we give an algorithm to compute a Gr¨obner basis for irreducible decomposition of algebraic varieties so as to find algebraic relations among coefficients of the damping function and the restoring function. We demonstrate the algorithm for polynomial Li′enard systems of degree 5, which are divided into 25 cases. We find all conditions of those coefficients for the polynomial Li′enard center in 13 cases and prove that the origin is not a center in the other 12 cases.
基金supported by the National Natural Science Foundation of China under Grant No.11701370。
文摘It is a fundamental problem to determine the equivalence of indexed differential polynomials in both computer algebra and differential geometry.However,in the literature,there are no general computational theories for this problem.The main reasons are that the ideal generated by the basic syzygies cannot be finitely generated,and it involves eliminations of dummy indices and functions.This paper solves the problem by extending Grobner basis theory.The authors first present a division of the set of elementary indexed differential monomials E■ into disjoint subsets,by defining an equivalence relation on E■ based on Leibniz expansions of monomials.The equivalence relation on E■also induces a division of a Grobner basis of basic syzygies into disjoint subsets.Furthermore,the authors prove that the dummy index numbers of the sim-monomials of the elements in each equivalence class of E■ have upper bounds,and use the upper bounds to construct fundamental restricted rings.Finally,the canonical form of an indexed differential polynomial proves to be the normal form with respect to a subset of the Grobner basis in the fundamental restricted ring.
文摘By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk＇s formula.
