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

 

Logarithmic derivatives of solutions to linear differential equations


Author: Christopher J. Hillar
Journal: Proc. Amer. Math. Soc. 132 (2004), 2693-2701
MSC (2000): Primary 34M15, 13P10; Secondary 34A26
Published electronically: April 21, 2004
MathSciNet review: 2054796
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Abstract: Given an ordinary differential field $K$ of characteristic zero, it is known that if $y$ and $1/y$ satisfy linear differential equations with coefficients in $K$, then $y'/y$ is algebraic over $K$. We present a new short proof of this fact using Gröbner basis techniques and give a direct method for finding a polynomial over $K$ that $y'/y$ satisfies. Moreover, we provide explicit degree bounds and extend the result to fields with positive characteristic. Finally, we give an application of our method to a class of nonlinear differential equations.


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Additional Information

Christopher J. Hillar
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: chillar@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07444-1
PII: S 0002-9939(04)07444-1
Keywords: Logarithmic derivative, linear differential equation, differential field, Gr\"{o}bner basis
Received by editor(s): August 19, 2002
Received by editor(s) in revised form: July 1, 2003
Published electronically: April 21, 2004
Additional Notes: This work is supported under a National Science Foundation Graduate Research Fellowship.
Communicated by: Michael Stillman
Article copyright: © Copyright 2004 American Mathematical Society