Rational solutions of first-order differential equations∗

We prove that degrees of rational solutions of an algebraic differential equation F (dw/dz,w, z) = 0 are bounded. For given F an upper bound for degrees can be determined explicitly. This implies that one can find all rational solutions by solving algebraic equations. Consider the differential equation F (w′, w, z) = 0, (w′ = dw/dz) (1) where F is a polynomial in three variables. Theorem 1 For every F there exists a constant C = C(F ) such that degree of every rational solution w of (1) does not exceed C. This statement is not true for differential equations of higher order. Indeed, all functions wn(z) = z n satisfy ( z w′ w )′ = 0. We will show that the bound for degree C(F ) can be determined effectively. So theoretically it is possible to find all rational solutions of (1) by ∗AMS subject Classification: 30D35 †Supported by NSF grant DMS 9500636