In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithm...In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.展开更多
In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is de?ned and its basic properties are proved. The...In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is de?ned and its basic properties are proved. The authors give a partial method to decide whether a Laurent differential chain A is Laurent regular. The decision for whether A is Laurent regular is reduced to the decision of whether a univariate differential chain A1 is Laurent regular. For a univariate differential chain A1,the authors ?rst give a criterion for whether A1 is Laurent regular in terms of its generic zeros and then give partial results on deciding whether A1 is Laurent regular.展开更多
Hilbert problem 15 requires to understand Schubert's book. In this book, there is a theorem in §23, about the relation of the tangent lines from a point and the singular points of cubed curves with cusp near ...Hilbert problem 15 requires to understand Schubert's book. In this book, there is a theorem in §23, about the relation of the tangent lines from a point and the singular points of cubed curves with cusp near a 3-multiple straight line, which was obtained by the so called main trunk numbers, while for these numbers, Schubert said that he obtained them by experiences. So essentially Schubert even did not give any hint for the proof of this theorem. In this paper, by using the concept of generic point in the framework of Van der Waerden and Weil on algebraic geometry, and realizing Ritt-Wu method on computer, the authors prove that this theorem of Schubert is completely right.展开更多
The Bertrand curves were first studied using a computer by Wu （1987）. The same problem was studied using an improved version of Ritt-Wu＇s decomposition algorithm by Chou and Gao （1993）. This paper investigates th...The Bertrand curves were first studied using a computer by Wu （1987）. The same problem was studied using an improved version of Ritt-Wu＇s decomposition algorithm by Chou and Gao （1993）. This paper investigates the same problem for pseudo null Bertrand curves in Minkowski 3-space E3 1.展开更多
This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods.They are relations of some special lines:1)For ...This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods.They are relations of some special lines:1)For a given point,all the curves containing this point are considered.For any such curve,there are five lines.Two of them are the tangent lines of the curve passing through the given point.The other three are the lines connecting the given point with the cusp,the inflexion point and the intersection point of the tangent line at the cusp and the inflexion line.2)For a given point,the curves whose tangent line at the cusp passes through this point are considered.For any such curve,there are four lines.Three of them are the tangent lines passing through this point and the other is the line connect the given point and the inflexion point.3)For a given point,the curves whose cusp,inflexion point and the given point are collinear are considered.For any such curve,there are five lines.Three of them are tangent lines passing through the given point.The other two are the lines connecting the given point with the cusp and the intersection point of the tangent line at the cusp and the inflexion line.展开更多
We know that the dimension for an irreducible ascending chain ASC is a crucial concept in Ritt-Wu’s constructive theory of algebraic geometry. We can also define the dimension for an arbitrary ascending chain similar...We know that the dimension for an irreducible ascending chain ASC is a crucial concept in Ritt-Wu’s constructive theory of algebraic geometry. We can also define the dimension for an arbitrary ascending chain similarly. But one may say that this definition has no geometric meaning. In this note, we shall show that the dimension of an展开更多
We extend the concept of the resolvent of a prime ideal to the concept of theresolvent of a general ideal with respect to a set of parameters and propose an algorithmto construct the generalized resolvents based on Wu...We extend the concept of the resolvent of a prime ideal to the concept of theresolvent of a general ideal with respect to a set of parameters and propose an algorithmto construct the generalized resolvents based on Wu-Rits’s zero decomposition algorithm.Our generalized algorithm has the following applications. (1) For a reducible variety V,we can find a direction on which V is projected birationally to an irreducible hypersurface.(2) We give a new algorithm to find a primitive element for a finite algebraic extensionof a field of characteristic zero. (3) We present a complete method of finding parametricequations for algebraic curves. (4) We give a method of solving a system of polynomialequations to any given precision.展开更多
文摘In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.
基金supported by NKRDPC under Grant No.2018YFA0306702the National Natural Science Foundation of China under Grant No.11688101
文摘In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is de?ned and its basic properties are proved. The authors give a partial method to decide whether a Laurent differential chain A is Laurent regular. The decision for whether A is Laurent regular is reduced to the decision of whether a univariate differential chain A1 is Laurent regular. For a univariate differential chain A1,the authors ?rst give a criterion for whether A1 is Laurent regular in terms of its generic zeros and then give partial results on deciding whether A1 is Laurent regular.
文摘Hilbert problem 15 requires to understand Schubert's book. In this book, there is a theorem in §23, about the relation of the tangent lines from a point and the singular points of cubed curves with cusp near a 3-multiple straight line, which was obtained by the so called main trunk numbers, while for these numbers, Schubert said that he obtained them by experiences. So essentially Schubert even did not give any hint for the proof of this theorem. In this paper, by using the concept of generic point in the framework of Van der Waerden and Weil on algebraic geometry, and realizing Ritt-Wu method on computer, the authors prove that this theorem of Schubert is completely right.
文摘The Bertrand curves were first studied using a computer by Wu （1987）. The same problem was studied using an improved version of Ritt-Wu＇s decomposition algorithm by Chou and Gao （1993）. This paper investigates the same problem for pseudo null Bertrand curves in Minkowski 3-space E3 1.
文摘This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods.They are relations of some special lines:1)For a given point,all the curves containing this point are considered.For any such curve,there are five lines.Two of them are the tangent lines of the curve passing through the given point.The other three are the lines connecting the given point with the cusp,the inflexion point and the intersection point of the tangent line at the cusp and the inflexion line.2)For a given point,the curves whose tangent line at the cusp passes through this point are considered.For any such curve,there are four lines.Three of them are the tangent lines passing through this point and the other is the line connect the given point and the inflexion point.3)For a given point,the curves whose cusp,inflexion point and the given point are collinear are considered.For any such curve,there are five lines.Three of them are tangent lines passing through the given point.The other two are the lines connecting the given point with the cusp and the intersection point of the tangent line at the cusp and the inflexion line.
基金The work was supported in part by the Grant CIZR-8702108 of National Natural Science Foundation of China.
文摘We know that the dimension for an irreducible ascending chain ASC is a crucial concept in Ritt-Wu’s constructive theory of algebraic geometry. We can also define the dimension for an arbitrary ascending chain similarly. But one may say that this definition has no geometric meaning. In this note, we shall show that the dimension of an
文摘We extend the concept of the resolvent of a prime ideal to the concept of theresolvent of a general ideal with respect to a set of parameters and propose an algorithmto construct the generalized resolvents based on Wu-Rits’s zero decomposition algorithm.Our generalized algorithm has the following applications. (1) For a reducible variety V,we can find a direction on which V is projected birationally to an irreducible hypersurface.(2) We give a new algorithm to find a primitive element for a finite algebraic extensionof a field of characteristic zero. (3) We present a complete method of finding parametricequations for algebraic curves. (4) We give a method of solving a system of polynomialequations to any given precision.
