Rational General Solutions of Trivariate Rational Differential Systems

We generalize the method of Ngô and Winkler (J Symbolic Comput 46:1173–1186, 2011) for finding rational general solutions of a plane rational differential system to the case of a trivariate rational differential system. We give necessary and sufficient conditions for the trivariate rational differential system to have a rational solution based on proper reparametrization of invariant algebraic space curves. In fact, the problem for computing a rational solution of the trivariate rational differential system can be reduced to finding a linear rational solution of an autonomous differential equation. We prove that the linear rational solvability of this autonomous differential equation does not depend on the choice of proper parametrizations of invariant algebraic space curves. In addition, two different rational solutions corresponding to the same invariant algebraic space curve are related by a shifting of the variable. We also present a criterion for a rational solution to be a rational general solution.

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