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On a topological description of solutions of complex differential equations. (English) Zbl 1082.34076
The authors introduce a new concept to study multi-valued solutions of differential equations. Let $G$ and $D$ be two domains in $\Bbb C$ and $f$ be a meromorphic function in $G$. Denote by $n(G, D):=n(G,D, f)$ the number of domains $E_{k}\subset G$ which $f$ maps conformally and one-to-one onto $D$. The number $n(G, D)$ coincides with the number of simple islands of covering surfaces over the domain $D$ in Ahlfor’s theory of covering surfaces and $n(G, D)$ can also be considered as a characteristics of the function $f$ in an arbitrary domain $G$ describing the global behavior of functions $f$. Finally, the authors derive an estimate on this number for a meromorphic solution $f$ of an equation of the form $$(w')^{m}+r_{1}{w'}^{m-1}+...+r_{m}=0,$$ where $r_{j}$, $j=1,\dots, m$, is an analytic function in the variables $z$ and $w$.

34M05Entire and meromorphic solutions (ODE)
32H04Meromorphic mappings on analytic spaces
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