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Let F be an irreducible differential polynomial over k(t) with k being an algebraically closed field of characteristic zero. The authors prove that F = 0 has rational general solutions if and only if the differential algebraic function field over k(t) associated to F is generated over k(t) by constants, i.e., the variety defined by F descends to a variety over k. As a consequence, the authors prove that if F is of first order and has movable singularities then F has only finitely many rational solutions. [ABSTRACT FROM AUTHOR]

We study the rational solutions of systems of first-order algebraic partial differential equations and relate them to those of an associated autonomous system. We also describe how rational general solutions of these systems are related, and provide an algorithm in some particular case concerning the dimension of the associated algebraic variety. Our results can be considered as a generalization of the approach by L. X. C. Ngô and F. Winkler on algebraic ordinary differential equations of order one, adapted to systems of first-order algebraic partial differential equations. [ABSTRACT FROM AUTHOR]

This paper generalizes the method of Ngô and Winkler (2010, 2011) for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation (AODE) to the case of a higher order AODE, provided a proper parametrization of its solution hypersurface. The authors reduce the problem of finding the rational general solution of a higher order AODE to finding the rational general solution of an associated system. The rational general solutions of the original AODE and its associated system are in computable 1-1 correspondence. The authors give necessary and sufficient conditions for the associated system to have a rational solution based on proper reparametrization of invariant algebraic space curves. The authors also relate invariant space curves to first integrals and characterize rationally solvable systems by rational first integrals. [ABSTRACT FROM AUTHOR]

We generalize the method of Ngô and Winkler (J Symbolic Comput 46:1173-1186, ) 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. [ABSTRACT FROM AUTHOR]

Abstract: We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE , we give an exact degree bound for its rational solutions, based on the connection between rational solutions of and rational parametrizations of the plane algebraic curve defined by . For a first order autonomous ODE, we further give a polynomial time algorithm for computing a rational general solution if it exists based on the computation of Laurent series solutions and Padé approximants. Experimental results show that the algorithm is quite efficient. [Copyright &y& Elsevier]