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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 and f be a meromorphic function in G. Denote by n(G,D):=n(G,D,f) the number of domains E k 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,,m, is an analytic function in the variables z and w.

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