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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
DOI: https://doi.org/10.1090/S0002-9939-04-07444-1
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.


References [Enhancements On Off] (What's this?)

  • 1. D. Cox, J. Little, and D. O'Shea, Using Algebraic Geometry, Springer-Verlag, New York, 1998. MR 99h:13033
  • 2. W. A. Harris, Jr. and Y. Sibuya, The $n$-th Roots of Solutions of Linear Ordinary Differential Equations, Proc. Amer. Math. Soc. 97 (1986), 207-211. MR 87m:12007
  • 3. W. A. Harris, Jr. and Y. Sibuya, The Reciprocals of Solutions of Linear Ordinary Differential Equations, Adv. in Math. 58 (1985), 119-132. MR 87j:34012
  • 4. S. Lang, Algebra, 3rd edition, Addison-Wesley Publishing Company, Reading, MA, 1993. MR 33:5416
  • 5. M. Saito, B. Sturmfels, and N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000. MR 2001i:13036
  • 6. M. F. Singer, Algebraic Relations among Solutions of Linear Differential Equations, Trans. Amer. Math. Soc. 295 (1986), 753-763. MR 87f:12015
  • 7. S. Sperber, Solutions of Differential Equations, Pacific Journal of Mathematics 124 (1986), 249-256. MR 87i:12017
  • 8. R. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, UK, 1999. MR 2000k:05026

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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: https://doi.org/10.1090/S0002-9939-04-07444-1
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

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