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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rational approximation to solutions of linear differential equations with algebraic coefficients


Author: Wolfgang M. Schmidt
Journal: Proc. Amer. Math. Soc. 53 (1975), 285-289
MSC: Primary 10F45; Secondary 10F25
MathSciNet review: 0387210
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Abstract: Let $ K$ be the field of formal series $ \alpha = {a_k}{t^k} + \cdots + {a_0} + {a_{ - 1}}{t^{ - 1}} + \cdots $ and let $ \vert\;\;\vert$ be the valuation with $ \vert\alpha \vert = {2^k}$ if $ {a_k} \ne 0$. Suppose $ \alpha \epsilon K$ satisfies an $ m{\text{th}}$ order linear differential equation whose coefficients are algebraic functions of $ t$. Then for $ \epsilon > 0$ there are only finitely many rational functions $ p(t)/q(t)$ with $ \vert\alpha - p(t)/q(t)\vert < \vert q(t){\vert^{ - 2 - 4m - \epsilon }}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0387210-2
PII: S 0002-9939(1975)0387210-2
Article copyright: © Copyright 1975 American Mathematical Society