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On algebraic solutions of linear differential equations with primitive unimodular Galois group

  • Felix Ulmer
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 539)

Abstract

The known algorithms for computing a liouvillian solution of an ordinary homogeneous linear differential equation L(y) = 0 use the fact that, if there is a liouvillian solution, then there is a solution z whose logarithmic derivative z"/z is algebraic over the field of coefficients. Their result is a minimal polynomial for z"/z. In this paper we show that, if there is no logarithmic derivative of a solution of small algebraic degree, then the solution z itself must be algebraic and the algebraic degree of z can be bounded. This can be used to improve algorithms computing liouvillian solutions and allows a direct computation of the minimal polynomial Q(ϑ) of z. In order to improve the computation of the minimal polynomial Q(ϑ), we get a criterion, in terms of the differential Galois group, from which the sparsity of Q(ϑ) can be derived.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Felix Ulmer
    • 1
  1. 1.Institut für Algorithmen und Kognitive SystemeUniversität Karlsruhe, Fakultät für InformatikKarlsruhe 1

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