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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

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