Rational Solutions of First Order Algebraic Ordinary Differential Equations

Let f(t,y,y’)=∑i=0nai(t,y)y’i=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y’)=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y’ but also on the coefficients of f viewed as the polynomial in t,y,y’.In this paper,the authors show that if f satisfies deg(f,y) <deg(f,y’) or■{deg(ai,y)-2)(n-i)}>0then the degree bound C only depends on the degrees of f in t,y,y’,and furthermore we present an explicit expression for C in terms of the degrees of f in t,y,y’.