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Oscillation of linear ordinary differential equations: on a theorem of A. Grigoriev. (English) Zbl 1131.34028

In 2001, A. Grigoriev [“Singular perturbations and zeros of abelian integrals”, Ph.D. thesis, Weizmann Inst. Sci., Rehovot] gave an extension of a theorem of de la Vallée Poussin to systems of first-order linear homogeneous differential equations with bounded polynomial coefficients. He proved that the number of isolated zeros of any linear combination of components of solutions of the system in the unit disk is bounded from above by a constant which only depends on the bounds of the coefficients, the degree of the polynomials and the dimension of the system. Here the author gives a new simpler proof of the results, and some extensions to Fuchsian systems.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
34A30 Linear ordinary differential equations and systems
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References:

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