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
Riemman metric tensor(Rmt)plays a significant role in deducing basic formulas and equations arising in differential geometry and(pseudo-)Riemannian manifolds.It is a fundamental and challenging problem to determine th...Riemman metric tensor(Rmt)plays a significant role in deducing basic formulas and equations arising in differential geometry and(pseudo-)Riemannian manifolds.It is a fundamental and challenging problem to determine the equivalence of indexed differential Rmt polynomials.This paper solves the problem by extending Gröbner basis theory and the previous work on the computational theory for indexed differentials.L-expansion of an indexed differential Rmt polynomial is defined.Then a decomposed form of the Gröbner basis of defining syzygies of the polynomial ring is presented,based on a partition of elementary indexed monomials.Meanwhile,the upper bound of the dummy index numbers of sim-monomials of the elements in each disjoint elementary indexed monomial subset is found.Finally,a DST-fundamental restricted ring is constructed,and the canonical form of a polynomial is confirmed to be the normal form with respect to the Gröbner basis in the DST-fundamental restricted ring.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
Spherical objects are widely used in target localization applications,and the existing sphere localization methods with cameras or total stations both have some limitations.A new high-precision sphere localization met...Spherical objects are widely used in target localization applications,and the existing sphere localization methods with cameras or total stations both have some limitations.A new high-precision sphere localization method with a theodolite is proposed in this paper.From the view point of the theodolite,the contour points of a sphere with a known radius are measured as latitude-longitude coordinates.It is observed that the center of the target sphere is located on a cylindrical surface constructed with the latitude-longitude coordinates,and therefore the latitude-longitude coordinates of at least three contour points can be used to construct a set of ternary quadratic equations.The Gröbner basis method is used to compute at most four real solutions of the sphere center coordinates.To distinguish the only meaningful solution from the other possible real solutions,a pre-processing of the measured longitude values is also proposed.The factors affecting the positioning accuracy of the sphere center are evaluated in simulation experiments,which are used to obtain an empirical estimation model of the positioning error.Real data experiments are also performed and the results show that the proposed method can achieve high localization precision.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 Nos.12371508 and 11701370。
文摘Riemman metric tensor(Rmt)plays a significant role in deducing basic formulas and equations arising in differential geometry and(pseudo-)Riemannian manifolds.It is a fundamental and challenging problem to determine the equivalence of indexed differential Rmt polynomials.This paper solves the problem by extending Gröbner basis theory and the previous work on the computational theory for indexed differentials.L-expansion of an indexed differential Rmt polynomial is defined.Then a decomposed form of the Gröbner basis of defining syzygies of the polynomial ring is presented,based on a partition of elementary indexed monomials.Meanwhile,the upper bound of the dummy index numbers of sim-monomials of the elements in each disjoint elementary indexed monomial subset is found.Finally,a DST-fundamental restricted ring is constructed,and the canonical form of a polynomial is confirmed to be the normal form with respect to the Gröbner basis in the DST-fundamental restricted ring.
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 in part by the National Natural Science Foundation of China under Grants 61703373,61873246,U1504604in part by the Key research project of Henan Province Universities under Grant 19A413014.
文摘Spherical objects are widely used in target localization applications,and the existing sphere localization methods with cameras or total stations both have some limitations.A new high-precision sphere localization method with a theodolite is proposed in this paper.From the view point of the theodolite,the contour points of a sphere with a known radius are measured as latitude-longitude coordinates.It is observed that the center of the target sphere is located on a cylindrical surface constructed with the latitude-longitude coordinates,and therefore the latitude-longitude coordinates of at least three contour points can be used to construct a set of ternary quadratic equations.The Gröbner basis method is used to compute at most four real solutions of the sphere center coordinates.To distinguish the only meaningful solution from the other possible real solutions,a pre-processing of the measured longitude values is also proposed.The factors affecting the positioning accuracy of the sphere center are evaluated in simulation experiments,which are used to obtain an empirical estimation model of the positioning error.Real data experiments are also performed and the results show that the proposed method can achieve high localization precision.
