Rational Solutions of First Order Algebraic Ordinary Differential Equations

05/04/2020

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by   Ruyong Feng, et al.

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Let f(t, y,y')=∑_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 { 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.