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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© 1991 Springer-Verlag Berlin Heidelberg
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Ulmer, F. (1991). On algebraic solutions of linear differential equations with primitive unimodular Galois group. In: Mattson, H.F., Mora, T., Rao, T.R.N. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1991. Lecture Notes in Computer Science, vol 539. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54522-0_132
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DOI: https://doi.org/10.1007/3-540-54522-0_132
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