Results 1  10
of
19
An introduction to commutative and noncommutative Gröbner bases
 Theoretical Computer Science
, 1994
"... In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1],[B2]). Since the end of the Seventies, Gröbner bases have been an essential tool in the development of computational ..."
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Cited by 71 (3 self)
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In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1],[B2]). Since the end of the Seventies, Gröbner bases have been an essential tool in the development of computational
Factoring and decomposing a class of linear functional systems, in "Linear Algebra Appl
"... Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, overdetermined, underdetermined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators ..."
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Cited by 28 (21 self)
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Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, overdetermined, underdetermined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, timedelay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M ′, where M (resp., M ′ ) is a module intrinsically associated with the linear functional system R y = 0 (resp., R ′ z = 0). These morphisms define applications sending solutions of the system R ′ z = 0 to solutions of R y = 0. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a noninjective endomorphism of the module M is equivalent to the existence of a nontrivial factorization R = R2 R1 of the system matrix R. The corresponding system can then be integrated “in cascade”. Under certain conditions, we also show that the system R y = 0 is equivalent to a system R ′ z = 0, where R ′ is a blocktriangular matrix of the same size as R. We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system R y = 0 to the integration of two independent systems R1 y1 = 0 and R2 y2 = 0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system R y = 0, i.e., they allow us to compute an equivalent system R ′ z = 0, where R ′ is a blockdiagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in a Maple package Morphisms based on the library OreModules.
String rewriting and Gröbner bases  a general approach to monoid and group rings
 Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita
, 1995
"... The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The tech ..."
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Cited by 15 (5 self)
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The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semiThue system. For certain presentations, including free groups and contextfree groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...
Noncommutative Computer Algebra for polynomial algebras: Gröbner bases, applications and implementation
, 2005
"... ..."
A combinatorial approach to involution and δregularity II: Structure analysis of polynomial modules with Pommaret bases
, 2002
"... Abstract Involutive bases are a special form of nonreduced Gröbner bases with additional combinatorial properties. Their origin lies in the JanetRiquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also nonco ..."
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Cited by 12 (3 self)
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Abstract Involutive bases are a special form of nonreduced Gröbner bases with additional combinatorial properties. Their origin lies in the JanetRiquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also noncommutative algebras like those generated by linear differential and difference operators or universal enveloping algebras of (finitedimensional) Lie algebras. A number of basic properties are derived and we provide concrete algorithms for their construction. Furthermore, we develop a theory for involutive bases with respect to semigroup orders (as they appear in local computations) and over coefficient rings, respectively. In both cases it turns out that generally only weak involutive bases exist. 1
Linear Control Systems over Ore Algebras: Effective Algorithms for the Computation of Parametrizations
 CDRom of the Workshop on TimeDelay Systems (TDS03), IFAC Workshop, INRIA Rocquencourt (France
, 2003
"... In this paper, we study linear control systems over Ore algebras. Within this mathematical framework... ..."
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Cited by 6 (3 self)
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In this paper, we study linear control systems over Ore algebras. Within this mathematical framework...
Introducing reduction to polycyclic group rings  a comparison of methods
, 1996
"... It is wellknown that for the integral group ring of a polycyclic group several decision problems are decidable. In this paper a technique to solve the membership problem for right ideals originating from Baumslag, Cannonito and Miller and studied by Sims is outlined. We want to analyze, how these d ..."
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Cited by 6 (3 self)
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It is wellknown that for the integral group ring of a polycyclic group several decision problems are decidable. In this paper a technique to solve the membership problem for right ideals originating from Baumslag, Cannonito and Miller and studied by Sims is outlined. We want to analyze, how these decision methods are related to Grobner bases. Therefore, we define effective reduction for group rings over Abelian groups, nilpotent groups and more general polycyclic groups. Using these reductions we present generalizations of Buchberger's Gröbner basis method by giving an appropriate definition of "Gröbner bases" in the respective setting and by characterizing them using concepts of saturation and spolynomials.
Intersection of ideals with non–commutative subalgebras
 In Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’06
, 2006
"... Abstract. Computation of an intersection of a left ideal with a subalgebra, which is not fully investigated until now, is important for different areas of mathematics. We present an algorithm for the computation of the preimage of a left ideal under a morphism of non–commutative GR–algebras, and sho ..."
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Cited by 2 (2 self)
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Abstract. Computation of an intersection of a left ideal with a subalgebra, which is not fully investigated until now, is important for different areas of mathematics. We present an algorithm for the computation of the preimage of a left ideal under a morphism of non–commutative GR–algebras, and show both its abilities and limitations. The main computational tools are the elimination of variables by means of Gröbner bases together with the constructive treatment of opposite algebras and the utilization of a special bimodule structure. Keywords: Non–commutative algebra, Groebner bases, elimination, intersection with subalgebra, preimage of ideal, homomorphism of algebras, restriction. 1.
Effective Gröbner structures
, 1997
"... Since Buchberger introduced the theory of Gröbner bases in 1965 it has become one of the most important tools in constructive algebra and, nowadays, it is the kernel of many algorithms in the theory of polynomial ideals and algebraic geometry. Motivated by the results in polynomial rings there have ..."
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Cited by 2 (1 self)
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Since Buchberger introduced the theory of Gröbner bases in 1965 it has become one of the most important tools in constructive algebra and, nowadays, it is the kernel of many algorithms in the theory of polynomial ideals and algebraic geometry. Motivated by the results in polynomial rings there have been investigated a lot of possibilities to generalise Buchberger's ideas to other types of rings. The perhaps most general concept, though it does not cover all extensions reported in the literature, is the extension to graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Gröbner bases it needs additional computability assumptions. The subject of this paper is the presentation of some classes of effective graded structures.