Computation of bases of free modules over the Weyl algebras

Quadrat, Alban; Robertz, Daniel

doi:10.1016/j.jsc.2007.06.005

Cited by 29 publications

(68 citation statements)

References 27 publications

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“…Using the recent extension of Stafford's main theorem to the rings D n (k) and D n (k) obtained in [9], we also generalize Stafford's theorems to the algebras D n (k) and D n (k). Finally, the algorithms we developed in the present paper for A n (Q) are implemented in the Stafford package ( [36]). …”

Section: If Rank D (M )mentioning

confidence: 99%

“…One of the main motivations for making Stafford's results constructive is the fact that they have system-theoretic interpretations within the algebraic analysis or D-modules approach ( [6,23,28,29]) (e.g., efficient generation of the set of autonomous elements ( [11,36]), computation of injective parametrizations ( [36]), Serre's reduction ( [7,15]), decomposability of the solution space ( [13])). These results apply to general linear PD systems which can be determined, overdetermined, or underdetermined depending on which result we consider.…”

Section: If Rank D (M )mentioning

confidence: 99%

“…We shortly recall a few results which will be useful in what follows. For more details and results, we refer to [11,13,36].…”

Section: Algebraic Analysismentioning

confidence: 99%

“…Constructive proofs of this result were studied in [21,25] and implemented in Macaulay2 ( [26]) and in Maple ( [36]). A consequence of Stafford's theorem is that every finitely generated projective (or, equivalently, stably free) left/right module over A n (k) of rank at least two is free ( [42]).…”

Section: Introductionmentioning

confidence: 99%

“…A consequence of Stafford's theorem is that every finitely generated projective (or, equivalently, stably free) left/right module over A n (k) of rank at least two is free ( [42]). This result was made constructive in [20] by following Stafford's original proof and in [36] by developing a more general one based on the concept of stable rank (Stafford's theorem asserting that the stable rank of A n (k) is two). This last algorithm was implemented in the Stafford package ( [36]) and used in mathematical systems theory ( [36]).…”

Section: Introductionmentioning

confidence: 99%

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A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Quadrat

Robertz

2014

Acta Appl Math

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To cite this version:Alban Quadrat, Daniel Robertz. A constructive study of the module structure of rings of partial differential operators. [ Abstract:The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two to two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r + 1 elements but no fewer. These results are implemented in the Stafford package for D = A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. Key-words:Weyl algebras, Stafford's theorems, linear systems of partial differential equations, D-modules, mathematical systems theory, constructive algebra, symbolic computation. A constructive study of the module structure of rings of partial differential operators 4

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Section: If Rank D (M )mentioning

confidence: 99%

Section: If Rank D (M )mentioning

confidence: 99%

“…We shortly recall a few results which will be useful in what follows. For more details and results, we refer to [11,13,36].…”

Section: Algebraic Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Quadrat

Robertz

2014

Acta Appl Math

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Effective Algebraic Analysis Approach to Linear Systems over Ore Algebras

Cluzeau¹,

Koutschan²,

Quadrat³

et al. 2020

Algebraic and Symbolic Computation Methods in Dynamical Systems

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The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra − based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras − and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OREALGEBRAIC-ANALYSIS, based on the Mathematica package HOLONOMICFUNCTIONS, is demonstrated.

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