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homalg  A metapackage for homological algebra
 J. Algebra Appl
"... Abstract. The central notion of this work is that of a functor between categories of finitely presented modules over socalled computable rings, i.e. rings R where one can algorithmically solve inhomogeneous linear equations with coefficients in R. The paper describes a way allowing one to realize s ..."
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Abstract. The central notion of this work is that of a functor between categories of finitely presented modules over socalled computable rings, i.e. rings R where one can algorithmically solve inhomogeneous linear equations with coefficients in R. The paper describes a way allowing one to realize such functors, e.g. HomR, ⊗R, Ext i R, Tor R i, as a mathematical object in a computer algebra system. Once this is achieved, one can compose and derive functors and even iterate this process without the need of any specific knowledge of these functors. These ideas are realized in the ring independent package homalg. It is designed to extend any computer algebra software implementing the arithmetics of a computable ring R, as soon as the latter contains algorithms to solve inhomogeneous linear equations with coefficients in R. Beside explaining how this suffices, the paper describes the nature of the extensions provided by homalg. 1.
Computing invariants of multidimensional linear systems on an abstract homological level
 Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto (Japan), 2006
"... Abstract — Methods from homological algebra [16] play a more and more important role in the study of multidimensional linear systems [15], [14], [6]. The use of modules allows an algebraic treatment of linear systems which is independent of their presentations by systems of equations. The type of li ..."
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Cited by 2 (2 self)
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Abstract — Methods from homological algebra [16] play a more and more important role in the study of multidimensional linear systems [15], [14], [6]. The use of modules allows an algebraic treatment of linear systems which is independent of their presentations by systems of equations. The type of linear system (ordinary/partial differential equations, timedelay systems, discrete systems...) is encoded in the (noncommutative) ring of (differential, shift,...) operators over which the modules are defined. In this framework, homological algebra gives very general information about the structural properties of linear systems. Homological algebra is a natural extension of the theory of modules over rings. The category of modules and their homomorphisms
Noether normalization guided by monomial cone decompositions
"... This paper explains the relevance of partitioning the set of standard monomials into cones for constructing a Noether normalization for an ideal in a polynomial ring. Such a decomposition of the complement of the corresponding initial ideal in the set of all monomials – also known as a Stanley decom ..."
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This paper explains the relevance of partitioning the set of standard monomials into cones for constructing a Noether normalization for an ideal in a polynomial ring. Such a decomposition of the complement of the corresponding initial ideal in the set of all monomials – also known as a Stanley decomposition – is constructed in the context of Janet bases, in order to come up with sparse coordinate changes which achieve Noether normal position for the given ideal.
Parc Orsay Université
, 2012
"... Symbolic methods for developing new domain decomposition algorithms ..."
Computer Science Journal of Moldova, vol.20, no.2(59), 2012 Computation of Difference Gröbner Bases
"... This paper is an updated and extended version of our note [1] (cf. also [2]). To compute difference Gröbner bases of ideals generated by linear polynomials we adopt to difference polynomial rings the involutive algorithm based on Janetlike division. The algorithm has been implemented in Maple in th ..."
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This paper is an updated and extended version of our note [1] (cf. also [2]). To compute difference Gröbner bases of ideals generated by linear polynomials we adopt to difference polynomial rings the involutive algorithm based on Janetlike division. The algorithm has been implemented in Maple in the form of the package LDA (Linear Difference Algebra) and we describe the main features of the package. Its applications are illustrated by generation of finite difference approximations to linear partial differential equations and by reduction of Feynman integrals. We also present the algorithm for an ideal generated by a finite set of nonlinear difference polynomials. If the algorithm terminates, then it constructs a Gröbner basis of the ideal. 1