The growth of solutions of algebraic differential equations
The paper is a brief survey of several results related to the following conjecture. Let \(f\) be a meromorphic solution of the algebraic differential equation: (1) \(P(z,f,f',\dots,f^{(n)})=0\), where \(P\) is a polynomial in all its arguments. Conjecture: If \(f(z)\) is a solution of (1) and \(T(r,f)\) denotes the Nevanlinna characteristic function of \(f(z)\), we have \(T(r,f)<a\exp_{n- 1}(br^c)\), \(0\leq r<\infty\), where \(a\), \(b\), \(c\) are positive constants and \(\exp_k(x)\) is the \(k\) times iterated exponential, i.e. \(\exp_0(x)= x\), \(\exp_1(x)= e^x\), \(\exp_k(x)= \exp_1(\exp_{k- 1}(x))\). The full solution of this conjecture is only known for first-order equations, i.e. \(P\) is a polynomial in \(f\), \(f'\) only. Then all meromorphic solutions of (1) have finite order. Some illustrative examples for second-order equations are given.
scientific article; zbMATH DE number 991898