Rational Solutions of High-Order Algebraic Ordinary Differential Equations

We consider algebraic ordinary differential equations (AODEs) and study their polynomial and rational solutions. A sufficient condition for an AODE to have a degree bound for its polynomial solutions is presented. An AODE satisfying this condition is called \emph{noncritical}. We prove that usual low order classes of AODEs are noncritical. For rational solutions, we determine a class of AODEs, which are called \emph{maximally comparable}, such that the poles of their rational solutions are recognizable from their coefficients. This generalizes a fact from linear AODEs, that the poles of their rational solutions are the zeros of the corresponding highest coefficient. An algorithm for determining all rational solutions, if there is any, of certain maximally comparable AODEs, which covers $78.54\%$ AODEs from a standard differential equations collection by Kamke, is presented

Rational Solutions of First-Order Algebraic Ordinary Difference Equations

We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees of its rational solutions, and thus derive a complete algorithm for computing corresponding rational solutions