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On singular points of solutions of linear differential systems with polynomial coefficients

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Abstract

We consider systems of linear ordinary differential equations containing m unknown functions of a single variable x. The coefficients of the systems are polynomials over a field k of characteristic 0. Each of the systems consists of m equations independent over k[x, d/dx]. The equations are of arbitrary orders. We propose a computer algebra algorithm that, given a system S of this form, constructs a polynomial d(x) ∈ k[x] \ {0} such that if S possesses a solution in \( \bar{k}{\left( {\left( {x - \alpha } \right)} \right)^m} \) for some \( \alpha \in \bar{k} \) and a component of this solution has a nonzero polar part, then d(α) = 0. In the case where k ⊆ ℂ and S possesses an analytic solution having a singularity of an arbitrary type (not necessarily a pole) at α, the equality d(α) = 0 is also satisfied.

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Authors and Affiliations

  1. Computing Centre of the Russian Academy of Science, Vavilov str., 40, 119333, GSP-1, Moscow, Russia

    S. A. Abramov & D. E. Khmelnov

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  1. S. A. Abramov
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  2. D. E. Khmelnov
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Correspondence to S. A. Abramov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 1, pp. 3–21, 2011/12.

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Abramov, S.A., Khmelnov, D.E. On singular points of solutions of linear differential systems with polynomial coefficients. J Math Sci 185, 347–359 (2012). https://doi.org/10.1007/s10958-012-0919-8

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  • DOI: https://doi.org/10.1007/s10958-012-0919-8

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