LIOUVILLIAN INTEGRABILITY OF POLYNOMIAL DIFFERENTIAL SYSTEMS

作者：Zhang, Xiang^*

来源：Transactions of the American Mathematical Society, 2016, 368(1): 607-620.

• Liouville integrability
• Darboux integrability
• Jacobian multiplier
• Galois group

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.

• 出版日期2016-1
• 单位上海交通大学