Meromorphic solutions of first-order algebraic differential equations (Q1109929)

Let K be the field of germs of meromorphic functions at \(\infty\), M be the field of functions which are meromorphic for \(r<| z| <\infty\) (r depends on the function). By \(K[t_ 1,...,t_ n]\), \(K(t_ 1,...,t_ n)\), respectively, we denote the ring of polynomials and field of rational functions in \(t_ 1,...,t_ n\) over K. Let \(y\in M\) be a solution of the differential equation \(F(y',y)=0\), \(F\in K[t_ 1,t_ 2]\). We study the order of the function y, i.e., the order of growth of the Nevanlinna characteristic T(r,y) as \(r\to \infty\).