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

The $ n$th roots of solutions of linear ordinary differential equations


Authors: William A. Harris and Yasutaka Sibuya
Journal: Proc. Amer. Math. Soc. 97 (1986), 207-211
MSC: Primary 12H05; Secondary 34A30
MathSciNet review: 835866
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Abstract: In this paper we shall prove the following theorem: Let $ K$ be a differential field of characteristic zero. Let $ \varphi $ and $ \psi $ be elements of a differential field extension of $ K$ such that (i) $ \varphi \ne 0$ and $ \psi \ne 0$; (ii) $ \varphi $ and $ \psi $ satisfy nontrivial linear differential equations with coefficients in $ K$, say, $ P(\varphi ) = 0$ and $ Q(\psi ) = 0$; (iii) $ \varphi = {\psi ^n}$ for some positive integer $ n$ such that $ n \geqslant {\text{ ord }}P$. Then the logarithmic derivatives of $ \varphi $ and $ \psi $ are algebraic over $ K$. (Note that $ \varphi '/\varphi = n(\psi '/\psi )$.)


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0835866-7
PII: S 0002-9939(1986)0835866-7
Article copyright: © Copyright 1986 American Mathematical Society