Rational Solutions of First Order Algebraic Ordinary Differential Equations

Let $f(t,y,y^{\mathrm{\prime }})={\sum }_{i=0}^d{a}_i(t,y)y^{\mathrm{\prime }i}=0$ 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 $f(t, y,y’)=0$isofdegreenotgreaterthan$C$. Examples show that this degree bound $C$ depends not only on the degrees of $f$ in $t,y,y^{\mathrm{\prime }}$ but also on the coefficients of $f$ viewed as polynomial in $t,y,y^{\mathrm{\prime }}$. In this paper, we show that if $$\underset{i=0}{\overset{d}{\max }}\{\mathrm{d}\mathrm{e}\mathrm{g}(a_i,y)-2(d-i)\}>0$$ 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 $f$.