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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Algebraic relations among solutions of linear differential equations


Author: Michael F. Singer
Journal: Trans. Amer. Math. Soc. 295 (1986), 753-763
MSC: Primary 12H05; Secondary 34A20
MathSciNet review: 833707
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Abstract: Using power series methods, Harris and Sibuya [3,4] recently showed that if $ k$ is an ordinary differential field of characteristic zero and $ y \ne 0$ is an element of a differential extension of $ k$ such that $ y$ and $ 1/y$ satisfy linear differential equations with coefficients in $ k$, then $ y\prime/y$ is algebraic over $ k$. Using differential galois theory, we generalize this and characterize those polynomial relations among solutions of linear differential equations that force these solutions to have algebraic logarithmic derivatives. We also show that if $ f$ is an algebraic function of genus $ \geq 1$ and if $ y$ and $ f(y)$ or $ y$ and $ {e^{\int y}}$ satisfy linear differential equations, then $ y$ is an algebraic function.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0833707-X
PII: S 0002-9947(1986)0833707-X
Keywords: Linear differential equations, Picard-Vessiot theory, linear algebraic groups
Article copyright: © Copyright 1986 American Mathematical Society