Higher-Order Linear Differential Systems with Truncated Coefficients
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  package ISOLDE .
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- 2.Barkatou, M.A.: A rational version of Moser’s Algorithm. In: ISSAC 1995 Proceedings, pp. 297–302. ACM Press, New York (1995)Google Scholar
- 4.Barkatou, M.A., Cluzeau, T., El Bacha, C.: Algorithms for regular solutions of higher-order linear differential systems. In: Johnson, J.R., Park, H., Kaltofen, E. (eds.) ISSAC 2009 Proceedings, pp. 7–14. ACM Press, New York (2009)Google Scholar
- 6.Barkatou, M.A., El Bacha, C., Pflügel, E.: Simultaneously row- and column-reduced higher-order linear differential systems. In: Koepf, W. (ed.) ISSAC 2010 Proceedings, pp. 45–52. ACM Press, New York (2010)Google Scholar
- 8.Barkatou, M.A., Pflügel, E.: On the equivalence problem of linear differential systems and its application for factoring completely reducible systems. In: Gloor, O. (ed.) ISSAC 1998 Proceedings, pp. 268–275. ACM Press, New York (1998)Google Scholar
- 9.Barkatou, M.A., Pflügel, E.: Computing super-irreducible forms of systems of linear differential equations via Moser-reduction: A new approach. In: Dongming, W. (ed.) ISSAC 2007 Proceedings, pp. 1–8. ACM Press, New York (2007)Google Scholar
- 11.Barkatou, M.A., Pflügel, E.: The ISOLDE package. A SourceForge Open Source project (2006), http://isolde.sourceforge.net
- 14.Lutz, D.A., Schäfke, R.: On the identification and stability of formal invariants for singular differential equations. Linear Algebra and Its Applications 72, 1–46 (1985)Google Scholar
- 17.Maple online help: http://www.maplesoft.com/support/help/