Rational Solutions of First Order Algebraic Ordinary Differential Equations

Let f(t,y,y′)=∑i=0nai(t,y)y′i=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(t,y,{y^\prime}) = \sum\nolimits_{i = 0}^n {{a_i}(t,y){y^{\prime i}} = 0} $$\end{document} 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