Rational Solutions of First Order Algebraic Ordinary Differential Equations

TL;DR

This paper investigates rational solutions of first order algebraic ODEs with polynomial coefficients, establishing conditions under which the degree bound of solutions depends solely on the degrees of the polynomial coefficients, and provides explicit formulas for this bound.

Contribution

The paper identifies specific conditions on the polynomial coefficients that ensure the degree bound depends only on the degrees in variables, and derives explicit formulas for this bound.

Findings

Degree bound depends only on degrees of f in t,y,y' under certain conditions.

Explicit expression for the degree bound C is provided.

Conditions include deg(f,y)<deg(f,y') or a specific inequality involving degrees of coefficients.

Abstract

Let $f(t,y,y')=\sum_{i=0}^n a_i(t,y)y'^i=0$ 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, we show that if $f$ satisfies $deg(f,y)<deg(f,y')$ 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 $t,y,y'$, and furthermore we present an explicit expression for $C$ in terms of the degrees of $f$ in $t,y,y'$.