Properties of Rational General Solutions for First Order Multivariate Autonomous Rational Differential Systems

This paper studies the properties of rational general solutions for first order multivariate autonomous rational differential systems. We obtain a necessary and sufficient conditions for existence of rational general solution of first order multivariate autonomous rational differential system if the degree bound of rational solutions of this system is given. Due to the problem for computing a rational solution of the multivariate rational differential system can be reduced to finding a linear rational solution of an autonomous differential equation, we also prove that the linear rational solvability of the resulting 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 proved to be related by a shifting of the variable.