Abstract
Rational approximations of Padé and Padé type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be “better” than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the “Roth's theorem” holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions.
Keywords: Roth theorem, Schmidt theorem, Picard-Vessiot extensions
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Selected References
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- Chudnovsky G. V. Rational approximations to linear forms of exponentials and binomials. Proc Natl Acad Sci U S A. 1983 May;80(10):3138–3141. doi: 10.1073/pnas.80.10.3138. [DOI] [PMC free article] [PubMed] [Google Scholar]