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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
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