多项式代数及其应用
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摘要
B．Buchberger在域上的多项式环的单项式的集合中引入项序，并利用S-多项式，给出了一种算法，使得对多项式环中的任意给定的理想，从它的一组生成元，可计算得到一组被称为简化Gr(?)bner基的生成元，它具有“唯一性”的良好性质。所以利用Gr(?)bner基可以解决与理想相关的许多问题。此后，Gr(?)bner基的应用研究得到了迅速发展。Gr(?)bner基的应用研究包括代数方程组求解，多项式的因子分解，素理想的检验，代数流形的分解，纠错码中循环码和代数几何码的译码，密码学中高维线性递归阵列的分析与综合，多维系统理论，信号处理和求解整数规划等诸多领域。
Morita对偶理论起源于数域上的向量空间的对偶空间理论。许多代数学家均从事Morita对偶理论的研究。他们研究环扩张，Noether环，序列环，自同态环等的对偶性及自对偶。上个世纪九十年代，C．Menini和A．D．Rio在分次环中引入分次Morita对偶的概念，得到了与Morita对偶相类似的刻划。
本文研究了Gr(?)bner基理论在求矩阵极小多项式，判定矩阵可逆性和求逆矩阵等方面的应用，并讨论了多项式环的分次自对偶问题。具体内容如下：
·证明了域上的所有块循环矩阵组成的环同构于其上的多元多项式环的一个商环。因此，将求域上块循环矩阵的极小多项式转化为求一个环同态的核的简化Gr(?)bner基，从而给出了求块循环矩阵的极小多项式的算法。
·给出了域上的块循环矩阵可逆性的充要条件及其逆矩阵的算法。
·给出了四元素可除代数上的块循环矩阵可逆性的充要条件及其逆矩阵的两种算法。
·给出了准确计算域上有限群的群代数上的多项式环的理想的Gr(?)bner基的算法。
·定义了整代数线性规划，并给出了求解它的算法。
·给出了求域上有限群的群代数上的块循环矩阵的极小多项式的算法，
西安电子科技大学博士学位论文:多项式代数及其应用
奇异性判别法及其逆矩阵的求法.
·给出了域上有限群的群代数上的块对称循环矩阵的奇异性判别法及其
逆矩阵求法.
，定义了域上有限群的群代数上的混合块矩阵，并给出了它的可逆性的
判别法及其逆矩阵的求法.
·定义了环的分次三角扩张和分次平凡扩张，并讨论了他们有分次Morita
对偶与其初始子环有Morita对偶之间的关系.
·讨论了余生成子环上的多元多项式环作为不同群的分次环的分次自对
偶问题，并证明了有自对偶的环的一种特殊的分次三角扩张有分次自
对偶.
B. Buchberger introduced the concepts of term orders in the set of all power products and presented an algorithm for finding a special generators, a reduced Grobner basis which is unique for a given term order, of a given ideal of a polynomial ring over a field from any generators of the ideal by means of S-polynomials. So many problems on ideals can be solved by the theory of the Grobner basis. Since 1980s, many mathematicians have been engaged in studying the applications of the Grobner basis such as solving the system of algebraic equations, factoring polynomials, testing primary ideals, factoring algebraic manifolds, decoding circular codes in corrected codes and algebraically geometrical codes, analyzing and synthesizing high dimensional linear recurring arrays in cryptology, dealing with multidimensional systematic theory, signaling, solving integer programming and so on.
The theory of Morita duality comes from the theory of the dual space of a linear space, which was proposed by K. Morita and G. Azumaya in 1950s. Since then, the theory of Morita duality has become one of the important fields of Ring Theory and Module Theory. Many algebraists are engaged in studying this theory. They study the duality and selfduality of rings such as extensions of a ring with a Morita duality(or selfduality), Noether rings, series rings, endomorphism rings and so on. C. Menini and A. D. Rio introduced a graded Morita duality into Graded Ring Theory and obtained the similar characterizations of the Morita duality.
The aim of this paper is to study the applications of Grobner bases in finding the minimal polynomial of a given matrix and its inverse if it is nonsingular and to discuss selfdualities over a polynomial ring. The main results are listed in the following:
It is proved that the ring consisting of all block circulant matrices over a field is isomorphic to a factor ring of a polynomial ring in multivariables over the same field. So finding the minimal polynomial of a block circulant matrix is transformed into computing the reduced Grobner basis of a kernel of a ring homomorphism. Hence an algorithm for the minimal polynomial of a block circulant matrix over the field is presented.
A sufficient and necessary condition for determining singularity of a block circulant matrix over a field is given, and an algorithm for the inverse of a nonsingular block circulant matrix over the field is presented.
A sufficient and necessary condition for determining singularity of a block circulant matrix over a quaternion division algebra over a field is given, and two
algorithms for the inverse of a nonsingular block circulant matrix over the quaternion division algebra are presented.
An algorithm for Grobner basis for an ideal of a polynomial ring over a group algebra of a finite group over a field is given.
An integral algebraic linear programming is defined and an algorithm for solving the programming is given.
Algorithms for the minimal polynomial and the inverse of a given block circulant matrix over a group algebra of a finite group over a field are presented, and a method of determining singularity of this block circulant matrix is given.
Algorithms for the minimal polynomial and the inverse of a given block symmetric circulant matrix over a group algebra of a finite group over a field are presented. and a method of determining singularity of this block symmetric circulant matrix is given.
The definition of a mixed block matrix is given, and an algorithm for the inverse of a given mixed block matrix over a group algebra of a finite group over a field is presented, and a method of determining singularity of this mixed block matrix is given.
Graded triangular extensions and graded trivial extensions over a ring are defined respectively, and the relation between them with grade Morita dualities and their initial subrings with Morita dualities is discussed.
The graded selfduality of a polynomial ring in multivariables over a cogenerator ring as a graded ring of type different group i
Morita对偶理论起源于数域上的向量空间的对偶空间理论。许多代数学家均从事Morita对偶理论的研究。他们研究环扩张，Noether环，序列环，自同态环等的对偶性及自对偶。上个世纪九十年代，C．Menini和A．D．Rio在分次环中引入分次Morita对偶的概念，得到了与Morita对偶相类似的刻划。
本文研究了Gr(?)bner基理论在求矩阵极小多项式，判定矩阵可逆性和求逆矩阵等方面的应用，并讨论了多项式环的分次自对偶问题。具体内容如下：
·证明了域上的所有块循环矩阵组成的环同构于其上的多元多项式环的一个商环。因此，将求域上块循环矩阵的极小多项式转化为求一个环同态的核的简化Gr(?)bner基，从而给出了求块循环矩阵的极小多项式的算法。
·给出了域上的块循环矩阵可逆性的充要条件及其逆矩阵的算法。
·给出了四元素可除代数上的块循环矩阵可逆性的充要条件及其逆矩阵的两种算法。
·给出了准确计算域上有限群的群代数上的多项式环的理想的Gr(?)bner基的算法。
·定义了整代数线性规划，并给出了求解它的算法。
·给出了求域上有限群的群代数上的块循环矩阵的极小多项式的算法，
西安电子科技大学博士学位论文:多项式代数及其应用
奇异性判别法及其逆矩阵的求法.
·给出了域上有限群的群代数上的块对称循环矩阵的奇异性判别法及其
逆矩阵求法.
，定义了域上有限群的群代数上的混合块矩阵，并给出了它的可逆性的
判别法及其逆矩阵的求法.
·定义了环的分次三角扩张和分次平凡扩张，并讨论了他们有分次Morita
对偶与其初始子环有Morita对偶之间的关系.
·讨论了余生成子环上的多元多项式环作为不同群的分次环的分次自对
偶问题，并证明了有自对偶的环的一种特殊的分次三角扩张有分次自
对偶.
B. Buchberger introduced the concepts of term orders in the set of all power products and presented an algorithm for finding a special generators, a reduced Grobner basis which is unique for a given term order, of a given ideal of a polynomial ring over a field from any generators of the ideal by means of S-polynomials. So many problems on ideals can be solved by the theory of the Grobner basis. Since 1980s, many mathematicians have been engaged in studying the applications of the Grobner basis such as solving the system of algebraic equations, factoring polynomials, testing primary ideals, factoring algebraic manifolds, decoding circular codes in corrected codes and algebraically geometrical codes, analyzing and synthesizing high dimensional linear recurring arrays in cryptology, dealing with multidimensional systematic theory, signaling, solving integer programming and so on.
The theory of Morita duality comes from the theory of the dual space of a linear space, which was proposed by K. Morita and G. Azumaya in 1950s. Since then, the theory of Morita duality has become one of the important fields of Ring Theory and Module Theory. Many algebraists are engaged in studying this theory. They study the duality and selfduality of rings such as extensions of a ring with a Morita duality(or selfduality), Noether rings, series rings, endomorphism rings and so on. C. Menini and A. D. Rio introduced a graded Morita duality into Graded Ring Theory and obtained the similar characterizations of the Morita duality.
The aim of this paper is to study the applications of Grobner bases in finding the minimal polynomial of a given matrix and its inverse if it is nonsingular and to discuss selfdualities over a polynomial ring. The main results are listed in the following:
It is proved that the ring consisting of all block circulant matrices over a field is isomorphic to a factor ring of a polynomial ring in multivariables over the same field. So finding the minimal polynomial of a block circulant matrix is transformed into computing the reduced Grobner basis of a kernel of a ring homomorphism. Hence an algorithm for the minimal polynomial of a block circulant matrix over the field is presented.
A sufficient and necessary condition for determining singularity of a block circulant matrix over a field is given, and an algorithm for the inverse of a nonsingular block circulant matrix over the field is presented.
A sufficient and necessary condition for determining singularity of a block circulant matrix over a quaternion division algebra over a field is given, and two
algorithms for the inverse of a nonsingular block circulant matrix over the quaternion division algebra are presented.
An algorithm for Grobner basis for an ideal of a polynomial ring over a group algebra of a finite group over a field is given.
An integral algebraic linear programming is defined and an algorithm for solving the programming is given.
Algorithms for the minimal polynomial and the inverse of a given block circulant matrix over a group algebra of a finite group over a field are presented, and a method of determining singularity of this block circulant matrix is given.
Algorithms for the minimal polynomial and the inverse of a given block symmetric circulant matrix over a group algebra of a finite group over a field are presented. and a method of determining singularity of this block symmetric circulant matrix is given.
The definition of a mixed block matrix is given, and an algorithm for the inverse of a given mixed block matrix over a group algebra of a finite group over a field is presented, and a method of determining singularity of this mixed block matrix is given.
Graded triangular extensions and graded trivial extensions over a ring are defined respectively, and the relation between them with grade Morita dualities and their initial subrings with Morita dualities is discussed.
The graded selfduality of a polynomial ring in multivariables over a cogenerator ring as a graded ring of type different group i
引文
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[2] A. Campillo, P. Gimenez. Syzygies of affine toric varieties. J. of Algebra. 2000, 225:142-161.
[3] A. D. Rio. Graded rings and equivalences of categories. Comm. in Algebra. 1991,19(3): 997-1012.
[4] A. D. Rio. Categorical methods in graded ring theory. Publications Matematiques. 1992, 36: 489-531.
[5] A. Jensen, S. Jondrup. Classical Quotient rings of group graded rings. Comm. in Algebra. 1992, 20(10): 2923-2936.
[6] A. V. Jategaonkar. Morita duality and Noetherian rings. J. of Algebra. 1981, 69:358-371.
[7] Bhubaneswar Mishra. Algorithmic Algebra. New York: Springer-Verlag, 2001.
[8] B. J. Müller. Linear compactness and Morita duality. J. of Algebra. 1970, 16: 60-66.
[9] B. J. Müller. Morita duality-a survey, Abelian gropu and modules. Proc. Conf.Udine/Italy 1984, CISM Courses Lecture. 1984, 287: 395-414.
[10] B. J. Miiller. On Morita duality. Canad. J. Mathematics. 1969, 21: 1338-1347.
[11] B. J. Miiller. Linear compactness and Morita duality. J. of algebra. 1970, 16: 60-66.
[12] B. L. Osofsky. A generalization of quasi-Frobenius rings. 3. of Algebra. 1966, 4. 373-387.
[13] Bo StenstrSm. Rings of Quotients. New York: Springer-Verlag, 1975.
[14] C. Cohen, S. Montgomery. Group-graded rings, smash products, and group actions. Transactions of American Mathematical Society. 1984, 282: 237-258.
[15] Chony-Yun Chao. A remark on symmetric circulant matrices. Linear Algebra and its Applications. 1988, 103: 133-148.
[16] C. Leary Bel. Generalized inverse of circulant and generalized circulant matrices. Linear Algebra and its Applications. 1981, 39: 133-142.
[17] C. Menini. Finitely graded rings, Morita duality and self-injectivity. Comm. in Algebra. 1987, 19: 1779-1797.
[18] C. Menini, A. D. Rio. Morita duality and graded rings. Comm. in Algebra. 1991, 19 1765-1795.
[19] C. Menini, C. Nǎstǎsescu. When is R-gr equivalent to the category of modules? J.of Pure and Applied Algebra. 1988, 51: 277-291.
[20] C. Menini, C. Nǎstǎsescu. Gr-simple modules and Gr-Jacobson radical: Applications
(Ⅰ), (Ⅱ). Bull. Math. De la Soc. Sci. Math. De Roumanie Tome. 1990, 34(82): 25-36, 125-133.
[21] C. Nǎstǎsescu. Some constructions over graded rings: Applications. J. of Algebra. 1989, 120: 119-138.
[22] C. Nǎstǎsescu, F. Van Oystaeyen. Graded Ring Theory. Amsterdam: Mathematical Library, North-Holland, 1980.
[23] D. Bersimas, G. Perakis and S. Tayur. A new algebraic geometry algorithm for integer programming. Management Science. 2000, 46(7): 999-1008.
[24] D. G. Northcott. Injective envelopes and inverse polynomials. J. London Mathematical Society. 1974, (2)8: 190-196.
[25] D. Quin. Group-graded rings and duality. Transactions of American Mathematical Society. 1985, 292: 155-167.
[26] F. L. Sandomierski. Linearly compact modules and local Morita duality. Proc. Conf. Ring Thery Utach, 1971, New York, London (Academic Press, 1972): 333-346.
[27] F. W. Anderson, K. R. Fuller. Rings and Categories of Modules. New York: Springer-Verlag, 1992.
[28] G. Azumaya. A duality theory for injective modules. American Journal of Mathematics. 1959, 81: 249-278.
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