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On transcendental solutions of diophantine equations. (Russian) Zbl 0665.30031
The author announces a rather general theorem of Picard type for the holomorphic mappings of complex spaces. The complete formulation is somewhat complicated, therefore we formulate a particular case only. We say that the polynomial $$p=p(x,y)\in C[x,y]$$ is of general type if for every $$c\in C\setminus E(p)$$, card $$E<\infty$$, all irreducible components of the curve $$\{(x,y)\in C^ 2: p(x,y)=c\}$$ are hyperbolic.
Theorem. Let p be the polynomial of general type and let the functions f, g be holomorphic in $$\{$$ z: $$0<| z| <1\}$$ and at most one of them has an essential singularity in zero. If the function p(f,g) does not have an essential singularity in zero, then p(f,g)$$\equiv const\in E(p)$$.
Reviewer: I.V.Ostrovskij

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables 30D30 Meromorphic functions of one complex variable, general theory