Rational Solutions of First Order Algebraic Ordinary Differential Equations

Let $f(t, y,y')=\sum_{i=0}^d a_i(t, y)y'^i=0$ be a first order ordinary differential equation with polynomial coefficients. Eremenko in 1999 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 polynomial in $t,y,y'$. In this paper, we show that if $$\max_{i=0}^d \{{\rm deg}(a_i,y)-2(d-i)\}>0$$ then the degree bound $C$ only depends on the degrees of $f$, and furthermore we present an explicit expression for $C$ in terms of the degrees of $f$.