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Liouvillian integrability of polynomial differential systems


Author: Xiang Zhang
Journal: Trans. Amer. Math. Soc. 368 (2016), 607-620
MSC (2010): Primary 34A34, 37C10, 34C14, 37G05
DOI: https://doi.org/10.1090/S0002-9947-2014-06387-3
Published electronically: November 12, 2014
MathSciNet review: 3413876
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Abstract: 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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Xiang Zhang
Affiliation: Department of Mathematics and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email: xzhang@sjtu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06387-3
Keywords: Liouville integrability, Darboux integrability, Jacobian multiplier, Galois group
Received by editor(s): June 27, 2013
Received by editor(s) in revised form: November 28, 2013
Published electronically: November 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society