On the equivalence problem of linear differential systems and its application for factoring completely reducible systems

Barkatou, Moulay A.; Pflügel, Eckhard

doi:10.1145/281508.281633

Cited by 16 publications

(31 citation statements)

References 11 publications

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“…The D-modules D/DLi, i = 1, 2 are isomorphic if and only if L1 ∼g L2. In particular, ∼g is an equivalence relation (see [1]). …”

Section: Transformationsmentioning

confidence: 99%

“…Projective equivalence is also an equivalence relation, see [1]. An implementation (for order 2) is given in [8] to decide if L1 ∼p L2, and if so, to find the projective equivalence (the r, r0, r1 in (1)).…”

Section: Transformationsmentioning

confidence: 99%

“…any (n) + · · · + a1y + a0y = 0. The problem of finding closed form solutions of L becomes easier if we can factor L as a product of lower order operators [2,7,1] or apply some other approach to reduce the order [9,14].…”

Section: Introductionmentioning

confidence: 99%

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2-descent for second order linear differential equations

Fang

Hoeij

2011

Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation

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Let L be a second order linear ordinary differential equation with coefficients in C(x). The goal in this paper is to reduce L to an equation that is easier to solve. The starting point is an irreducible L, of order two, and the goal is to decide if L is projectively equivalent to another equationL that is defined over a subfield C(f ) of C(x).This paper treats the case of 2-descent, which means reduction to a subfield with index [C(x) : C(f )] = 2. Although the mathematics has already been treated in other papers, a complete implementation could not be given because it involved a step for which we do not have a complete implementation. The contribution of this paper is to give an approach that is fully implementable [5]. Examples illustrate that this algorithm is very useful for finding closed form solutions (2-descent, if it exists, reduces the number of true singularities from n to at most n/2 + 2).

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“…The D-modules D/DLi, i = 1, 2 are isomorphic if and only if L1 ∼g L2. In particular, ∼g is an equivalence relation (see [1]). …”

Section: Transformationsmentioning

confidence: 99%

Section: Transformationsmentioning

confidence: 99%

See 1 more Smart Citation

2-descent for second order linear differential equations

Fang

Hoeij

2011

Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation

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“…(ii) It follows from the proof of (i) that no sequence (a i ) exists for the system S of the form (8). However, if Ω is a finite signal then any sequence (a l ) can be used for the system Ω * S because neither Ω * S nor its truncations have solutions in k((x)) \ {0}.…”

Section: Proof (I) Let S Be the Systemmentioning

confidence: 99%

Higher-Order Linear Differential Systems with Truncated Coefficients

Абрамов¹,

Barkatou²,

Pflügel³

2011

Computer Algebra in Scientific Computing

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Abstract. We 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].

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“…If M is known (i.e if the change of variables x → f is known), then the map from V (M ) to V (L) can be computed with existing algorithms [1], [6]. That means L √ B −→CEG L can be computed if we can find the change of variables.…”

Section: Transformationsmentioning

confidence: 99%

Finding all bessel type solutions for linear differential equations with rational function coefficients

Hoeij

Yuan

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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A linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bν (f ), Bν+1(f ). For second order equations, with rational function coefficients, f must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if f is a rational function. In this paper we extend this work to the square root case, resulting in a complete algorithm to find all Bessel type solutions.

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On the equivalence problem of linear differential systems and its application for factoring completely reducible systems

Cited by 16 publications

References 11 publications

2-descent for second order linear differential equations

2-descent for second order linear differential equations

Higher-Order Linear Differential Systems with Truncated Coefficients

Finding all bessel type solutions for linear differential equations with rational function coefficients

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