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- W1967511031 abstract "M.F. Singer (<italic>Liouvillian first integrals of differential equations</italic>, 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–dimensional polynomial differential system has <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n minus 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functionally independent Darboux Jacobian multipliers, then it has <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n minus 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functionally independent Liouvillian first integrals. Conversely if the system is Liouvillian integrable, then it has a Darboux Jacobian multiplier." @default.
- W1967511031 created "2016-06-24" @default.
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- W1967511031 date "2014-11-12" @default.
- W1967511031 modified "2023-09-26" @default.
- W1967511031 title "Liouvillian integrability of polynomial differential systems" @default.
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- W1967511031 doi "https://doi.org/10.1090/s0002-9947-2014-06387-3" @default.
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