Stability Phenomenon for Generalizations of Algebraic Differential Equations

Abstract

Certain stability properties for meromorphic solutions $w(z)=u(x,y)+iv(x,y)$ of partial differential equations of the form ${\sum }^m–t=0f_t(w^2{)}^{m–t}=0$ are considered. Here the coefficients $f_t$ are functions of $x,y$, of $u,v$ and the partial derivatives of $u,v$. Assuming that certain growth conditions for the coefficients $f_t$ are valid in the preimage under $w$ of five distinct complex values, we find growth estimates, in the whole complex plane, for the order $\rho (w)$ and the unintegrated Ahlfors-Shimizu characteristic $A(r,w)$.