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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Liouvillian integrability of polynomial differential systems
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by Xiang Zhang PDF
Trans. Amer. Math. Soc. 368 (2016), 607-620 Request permission


M.F. Singer (Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673–688) proved the equivalence between Liouvillian integrability and Darboux integrability for two dimensional polynomial differential systems. In this paper we will extend Singer’s result to any finite dimensional polynomial differential systems. We prove that if an $n$–dimensional polynomial differential system has $n-1$ functionally independent Darboux Jacobian multipliers, then it has $n-1$ functionally independent Liouvillian first integrals. Conversely if the system is Liouvillian integrable, then it has a Darboux Jacobian multiplier.
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Additional Information
  • Xiang Zhang
  • Affiliation: Department of Mathematics and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
  • Email:
  • Received by editor(s): June 27, 2013
  • Received by editor(s) in revised form: November 28, 2013
  • Published electronically: November 12, 2014
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 607-620
  • MSC (2010): Primary 34A34, 37C10, 34C14, 37G05
  • DOI:
  • MathSciNet review: 3413876