This paper provides a fast algorithm for Grobnerbases of homogenous ideals of F[x, y] over a finite field F. We show that only the 8-polynomials of neighbor pairs of a strictly ordered finite homogenours generating se...This paper provides a fast algorithm for Grobnerbases of homogenous ideals of F[x, y] over a finite field F. We show that only the 8-polynomials of neighbor pairs of a strictly ordered finite homogenours generating set are needed in the computing of a Grobner base of the homogenous ideal. It reduces dramatically the number of unnecessary 5-polynomials that are processed. We also show that the computational complexity of our new algorithm is O（N^2）, where N is the maximum degree of the input generating polynomials. The new algorithm can be used to solve a problem of blind recognition of convolutional codes. This problem is a new generalization of the important problem of synthesis of a linear recurring sequence.展开更多
We give more efficient criteria to characterise prime ideal or primary ideal. Further, we obtain the necessary and sufficient conditions that an ideal is prime or primary in real field from the Grobner bases directly.
Rational Univariate Representation(RUR) of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively.In 1999,Roullier proposed an efficient algorithm to comp...Rational Univariate Representation(RUR) of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively.In 1999,Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals.In this paper,we will present a new algorithm to compute Polynomial Univariate Representation(PUR) of zero-dimensional ideals.The new algorithm is based on some interesting properties of Grbner basis.The new algorithm also provides a method for testing separating elements.展开更多
Ⅰ. DEFINITION OF MINIMAL CHARACTERISTIC BASISLet K be a field of characteristic zero, and X1,…, Xn be variables fixed in what follows. Let K{X1,…, Xn} be the set of differential polynomials (abbr. dpols ) in X1,…,Xn.
We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obn...We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.展开更多
GVW algorithm was given by Gao,Wang,and Volny in computing a Grbner bases for ideal in a polynomial ring,which is much faster and more simple than F5.In this paper,the authors generalize GVW algorithm and present an...GVW algorithm was given by Gao,Wang,and Volny in computing a Grbner bases for ideal in a polynomial ring,which is much faster and more simple than F5.In this paper,the authors generalize GVW algorithm and present an algorithm to compute a Grbner bases for ideal when the coefficient ring is a principal ideal domain.展开更多
Farr-Gao algorithm is a state-of-the-art algorithm for reduced Gr?bner bases of vanishing ideals of finite points, which has been implemented in Maple as a build-in command. This paper presents a two-dimensional impro...Farr-Gao algorithm is a state-of-the-art algorithm for reduced Gr?bner bases of vanishing ideals of finite points, which has been implemented in Maple as a build-in command. This paper presents a two-dimensional improvement for it that employs a preprocessing strategy for computing reduced Gr?bner bases associated with tower subsets of given point sets. Experimental results show that the preprocessed Farr-Gao algorithm is more efficient than the classical one.展开更多
Both original and twisted Schrdinger-Virasoro algebras, and also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim at determining t...Both original and twisted Schrdinger-Virasoro algebras, and also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim at determining the 2-cocycles of original deformative Schrdinger-Virasoro algebras.展开更多
基金the National Natural Science Foundation of China (Grant No. 60673082)Special Funds of Authors of Excellent Doctoral Dissertation in China (Grant No. 200084)
文摘This paper provides a fast algorithm for Grobnerbases of homogenous ideals of F[x, y] over a finite field F. We show that only the 8-polynomials of neighbor pairs of a strictly ordered finite homogenours generating set are needed in the computing of a Grobner base of the homogenous ideal. It reduces dramatically the number of unnecessary 5-polynomials that are processed. We also show that the computational complexity of our new algorithm is O（N^2）, where N is the maximum degree of the input generating polynomials. The new algorithm can be used to solve a problem of blind recognition of convolutional codes. This problem is a new generalization of the important problem of synthesis of a linear recurring sequence.
基金Supported by the National Natural Science Foundation of China (No. 11071062)Hunan provincial Natural Science Foundation of China (No. 10JJ3065)+1 种基金Scientific Research Fund of Hunan province education Department (No. 10A033)Hunan Provincial Degree and Education of Graduate Student Foundation (No. JG2009A017)
文摘We give more efficient criteria to characterise prime ideal or primary ideal. Further, we obtain the necessary and sufficient conditions that an ideal is prime or primary in real field from the Grobner bases directly.
基金supported by National Key Basic Research Project of China(Grant No. 2011CB302400)National Natural Science Foundation of China (Grant Nos. 10971217,60821002/F02)
文摘Rational Univariate Representation(RUR) of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively.In 1999,Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals.In this paper,we will present a new algorithm to compute Polynomial Univariate Representation(PUR) of zero-dimensional ideals.The new algorithm is based on some interesting properties of Grbner basis.The new algorithm also provides a method for testing separating elements.
文摘Ⅰ. DEFINITION OF MINIMAL CHARACTERISTIC BASISLet K be a field of characteristic zero, and X1,…, Xn be variables fixed in what follows. Let K{X1,…, Xn} be the set of differential polynomials (abbr. dpols ) in X1,…,Xn.
基金supported by the Danish Council for Independent Research(Grant No.DFF–4002-00367),supported by the Danish Council for Independent Research(Grant No.DFF–6108-00362)supported by the Research Council of Norway(Project No.280731)supported by IRCC Award grant 12IRAWD009 from IIT Bombay
文摘We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.
基金supported by the National Natural Science Foundation of China under Grant Nos.11071062,11271208Scientific Research Fund of Hunan Province Education Department under Grant Nos.10A033,12C0130
文摘GVW algorithm was given by Gao,Wang,and Volny in computing a Grbner bases for ideal in a polynomial ring,which is much faster and more simple than F5.In this paper,the authors generalize GVW algorithm and present an algorithm to compute a Grbner bases for ideal when the coefficient ring is a principal ideal domain.
基金supported by the National Natural Science Foundation of China under Grant Nos.11101185 and 11171133
文摘Farr-Gao algorithm is a state-of-the-art algorithm for reduced Gr?bner bases of vanishing ideals of finite points, which has been implemented in Maple as a build-in command. This paper presents a two-dimensional improvement for it that employs a preprocessing strategy for computing reduced Gr?bner bases associated with tower subsets of given point sets. Experimental results show that the preprocessed Farr-Gao algorithm is more efficient than the classical one.
基金the National Natural Science Foundation of China (Grant Nos. 10471091, 10671027)Foundation of Shanghai Education Committee (Grant No. 06FZ029)"One Hundred Talents Program" from University of Science and Technology of China
文摘Both original and twisted Schrdinger-Virasoro algebras, and also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim at determining the 2-cocycles of original deformative Schrdinger-Virasoro algebras.
