This paper will prove that f≡g(modI) iff N F(f)=N F(g) for f,g∈K[x,],obtain a basis for the K vector space K[x,]/I,give the method for finding a Grbner basis of intersection of the left ideals I and J.
Improved algorithm for Grbner basis is a new way to solve Grbner basis by adopting the locally analytic method,which is based on GrbnerNew algorithm The process consists of relegating the leading terms of generator of...Improved algorithm for Grbner basis is a new way to solve Grbner basis by adopting the locally analytic method,which is based on GrbnerNew algorithm The process consists of relegating the leading terms of generator of the polynomial in the idea according to correlated expressions of leading terms and then analyzing every category.If a polynomial can be reduced to a remainder polynomial by a polynomial in the idea,then it can be replaced by the remainder polynomial as generator In the solving process,local reduction and local puwer decrease are employed to prevent the number of middle terms from increasing too fast and the degrees of polynomial from being too high so as to reduce the amount of展开更多
We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is...We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.展开更多
Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investiga...Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.展开更多
Different from previous viewpoints,multivariate polynomial matrix Diophantine equations are studied from the perspective of modules in this paper,that is,regarding the columns of matrices as elements in modules.A nece...Different from previous viewpoints,multivariate polynomial matrix Diophantine equations are studied from the perspective of modules in this paper,that is,regarding the columns of matrices as elements in modules.A necessary and sufficient condition of the existence for the solution of equations is derived.Using powerful features and theoretical foundation of Gr?bner bases for modules,the problem for determining and computing the solution of matrix Diophantine equations can be solved.Meanwhile,the authors make use of the extension on modules for the GVW algorithm that is a signature-based Gr?bner basis algorithm as a powerful tool for the computation of Gr?bner basis for module and the representation coefficients problem directly related to the particular solution of equations.As a consequence,a complete algorithm for solving multivariate polynomial matrix Diophantine equations by the Gr?bner basis method is presented and has been implemented on the computer algebra system Maple.展开更多
Zero-dimensional valuation rings are one kind of non-Noetherian rings.This paper investigates properties of zero-dimensional valuation rings and prove that a finitely generated ideal over such a ring has a Grobner bas...Zero-dimensional valuation rings are one kind of non-Noetherian rings.This paper investigates properties of zero-dimensional valuation rings and prove that a finitely generated ideal over such a ring has a Grobner basis.The authors present an algorithm for computing a Gr?bner basis of a finitely generated ideal over it.Furthermore,an interesting example is also provided to explain the algorithm.展开更多
In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct t...In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gr¨obner-Shirshov basis for degenerate Ringel-Hall algebras of type F_4.展开更多
研究了NGP(nearly general stewart-gough platform)并联机构动平台位置与姿态变量之间的耦合关系,将9个变量中的6个用其余的3个表达出来,从而实现了位置变量和姿态变量的解耦.运用Gr bner基算法,得到了15个只含有其余3个变量的4次相容...研究了NGP(nearly general stewart-gough platform)并联机构动平台位置与姿态变量之间的耦合关系,将9个变量中的6个用其余的3个表达出来,从而实现了位置变量和姿态变量的解耦.运用Gr bner基算法,得到了15个只含有其余3个变量的4次相容方程.在此基础上,采用变量代换的方法消去其中的高次项,最终将NGP并联机构的运动学正解问题简化为求解一个一元20次的代数方程;这个方程是通过计算一个10阶行列式得出的,并且通过一个具体的算例验证了该方法的正确性.该方法适用于所有的NGP并联机构.展开更多
文摘This paper will prove that f≡g(modI) iff N F(f)=N F(g) for f,g∈K[x,],obtain a basis for the K vector space K[x,]/I,give the method for finding a Grbner basis of intersection of the left ideals I and J.
文摘Improved algorithm for Grbner basis is a new way to solve Grbner basis by adopting the locally analytic method,which is based on GrbnerNew algorithm The process consists of relegating the leading terms of generator of the polynomial in the idea according to correlated expressions of leading terms and then analyzing every category.If a polynomial can be reduced to a remainder polynomial by a polynomial in the idea,then it can be replaced by the remainder polynomial as generator In the solving process,local reduction and local puwer decrease are employed to prevent the number of middle terms from increasing too fast and the degrees of polynomial from being too high so as to reduce the amount of
文摘We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.
文摘Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.
基金supported by the National Natural Science Foundation of China under Grant No.12001030the CAS Key Project QYZDJ-SSW-SYS022the National Key Research and Development Project2020YFA0712300。
文摘Different from previous viewpoints,multivariate polynomial matrix Diophantine equations are studied from the perspective of modules in this paper,that is,regarding the columns of matrices as elements in modules.A necessary and sufficient condition of the existence for the solution of equations is derived.Using powerful features and theoretical foundation of Gr?bner bases for modules,the problem for determining and computing the solution of matrix Diophantine equations can be solved.Meanwhile,the authors make use of the extension on modules for the GVW algorithm that is a signature-based Gr?bner basis algorithm as a powerful tool for the computation of Gr?bner basis for module and the representation coefficients problem directly related to the particular solution of equations.As a consequence,a complete algorithm for solving multivariate polynomial matrix Diophantine equations by the Gr?bner basis method is presented and has been implemented on the computer algebra system Maple.
基金supported by the National Natural Science Foundation of China under Grant Nos.11871207and 11971161。
文摘Zero-dimensional valuation rings are one kind of non-Noetherian rings.This paper investigates properties of zero-dimensional valuation rings and prove that a finitely generated ideal over such a ring has a Grobner basis.The authors present an algorithm for computing a Gr?bner basis of a finitely generated ideal over it.Furthermore,an interesting example is also provided to explain the algorithm.
基金supported by the National Natural Science Foundation of China(Nos.11361056,11271043,11471186)
文摘In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gr¨obner-Shirshov basis for degenerate Ringel-Hall algebras of type F_4.
文摘研究了NGP(nearly general stewart-gough platform)并联机构动平台位置与姿态变量之间的耦合关系,将9个变量中的6个用其余的3个表达出来,从而实现了位置变量和姿态变量的解耦.运用Gr bner基算法,得到了15个只含有其余3个变量的4次相容方程.在此基础上,采用变量代换的方法消去其中的高次项,最终将NGP并联机构的运动学正解问题简化为求解一个一元20次的代数方程;这个方程是通过计算一个10阶行列式得出的,并且通过一个具体的算例验证了该方法的正确性.该方法适用于所有的NGP并联机构.
