Rational Solutions of First Order Algebraic Ordinary Differential Equations

Abstract

Let $f(t,y,y^{\prime })={\sum }_{i=0}^n{a}_i(t,y)y^{\prime i}=0$ 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 $f(t,y,y^{\prime })=0$ is of degree not greater than $C$. Examples show that this degree bound $C$ depends not only on the degrees of $f$ in $t,y,y^{\prime }$ but also on the coefficients of $f$ viewed as the polynomial in $t,y,y^{\prime }$. In this paper, we show that if $f$ satisfies $deg(f,y)<deg(f,y^{\prime })$ or ${\max }_{i=0}^n\{deg(a_i,y)-2(n-i)\}>0$ then the degree bound $C$ only depends on the degrees of $f$ in $t,y,y^{\prime }$, and furthermore we present an explicit expression for $C$ in terms of the degrees of $f$ in $t,y,y^{\prime }$.

arXiv admin note: substantial text overlap with arXiv:2005.01289