Rational liouvillian solution of algebraic ordinary differential equations of order one in genus zero
Abstract
We study a necessary and sufficient condition for having a rational liouvilian solution of the autonomous algebraic ordinary differential equation f y , y ' =0 , where f X , Y =0 defines a rational algebraic curve (genus zero) over complex field ∁ . This article based on three ideas: every rational algebraic curve has proper parametrization, pair X t , X ' ( t ) is a proper parametrization of a certain algebraic curve, and the condition of differential equation of order one y ' = f ( y ) has liouvillian solution over C .