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Sur la topologie des polynômes complexes. (Topology of complex polynomials). (French) Zbl 0597.32005
Let $$f: {\mathbb{C}}^{n+1}\to {\mathbb{C}}$$ be a polynomial map. The authors show that there is a finite set $$A\subset {\mathbb{C}}$$ such that the restricted map $$f: {\mathbb{C}}^{n+1}-f^{-1}(A)\to {\mathbb{C}}-A$$ is a $$C^{\infty}$$ fibration. Furthermore they show that $$f: {\mathbb{C}}^ 2\to {\mathbb{C}}$$ is a locally trivial $$C^{\infty}$$ fibration in a neighbourhood of $$z_ 0\in {\mathbb{C}}$$ if and only if the following conditions (1) and (2) are satisfied. (1) $$z_ 0$$ is not a critical value of f. (2) The Euler- Poincaré characteristic of the fiber $$f^{-1}(z_ 0)$$ is equal to that of the general fiber $$f^{-1}(z)$$.
Reviewer: S.Tajima

##### MSC:
 32A15 Entire functions of several complex variables 32S05 Local complex singularities 55R55 Fiberings with singularities in algebraic topology