Liouvillian integrability of polynomial differential systems

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 multiplier then it has n-1 functionally independent Liouvillian first integrals. Conversely if the system is Liouvillian integrable then it has a Darboux Jacobian multiplier.