Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

On super-irreducible forms of linear differential systems with rational function coefficients

AbstractConsider a system of n linear first-order differential equations (d/dx)y=A(x)y in which A(x) is an n×n matrix of rational functions over a subfield F of the field C of complex numbers and let Γ={α1,…,αd}⊂C be a set of conjugate singularities of this system, i.e., poles of A(x) which are roots in C of some irreducible polynomial p(x) in F[x]. We propose an algorithm for transforming the given system into an equivalent system over F(x) which is super-irreducible in each element α∈Γ. This algorithm does not require working in the algebraic extension F(Γ) that appears when one applies Hilali–Wazner's algorithm (Numer. Math. 50 (1987) 429) successively with the individual singularities α1,…,αd. The transformation matrix as well as the resulting system have their coefficients in F(x) and all the computations are performed in F[x]/(p) instead of the splitting field of p

Higher-Order linear differential systems with truncated coefficients

International audienceWe consider the following problem: given a linear differential system with formal Laurent series coefficients, we want to decide whether the system has non-zero Laurent series solutions, and find all such solutions if they exist. Let us also assume we need only a given positive integer number l of initial terms of these series solutions. How many initial terms of the coefficients of the original system should we use to construct what we need? Supposing that the series coefficients of the original systems are represented algorithmically, we show that these questions are undecidable in general. However, they are decidable in the scalar case and in the case when we know in advance that a given system has an invertible leading matrix. We use our results in order to improve some functionality of the Maple [17] package ISOLDE [11]

Computable Infinite Power Series in the Role of Coefficients of Linear Differential Systems

International audienceWe consider linear ordinary differential systems over a differential field of characteristic 0. We prove that testing unimodularity and computing the dimension of the solution space of an arbitrary system can be done algorithmically if and only if the zero testing problem in the ground differential field is algorithmically decidable. Moreover, we consider full-rank systems whose coefficients are computable power series and we show that, despite the fact that such a system has a basis of formal exponential-logarithmic solutions involving only computable series, there is no algorithm to construct such a basis

Solutions asymptotiques d'equations aux differences a coefficients polynomiaux

SIGLECNRS 17660 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

On algebraic simplifications of linear functional systems

Summary. In this paper, we show how to conjointly use module theory and constructive homological algebra to obtain general conditions for a matrix R of functional operators (e.g., differential/shift/time-delay operators) to be equivalent to a block-triangular or block-diagonal matrix R (i.e., conditions for the existence of unimodular matrices V and W satisfying that R = V R W). These results allow us to simplify the study of many linear functional systems − particularly differential time-delay systems − appearing in control theory and mathematical physics

Algorithme de Newton rationnel pour le calcul des solutions formelles d'une equation differentielle lineaire en un point singulier irregulier

SIGLECNRS 17660 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc