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On polynomials whose fibers are irreducible with no critical points. (English) Zbl 0803.13009
In this work examples are given of polynomials \(F \in \mathbb{C} [X,Y]\) without critical points, i.e. \(\mu (F) = \dim_ \mathbb{C} \mathbb{C} [X,Y]/ (F_ X, F_ Y) = 0\) and whose fibers \(C_ \lambda = F^{-1} (\lambda)\) are irreducible for all \(\lambda \in \mathbb{C}\).– Such kind of examples, whose existence was not previously known, are of great interest in the study of the topology of polynomial maps from \(\mathbb{C}^ 2\) to \(\mathbb{C}\), for instance, to understand the topology at infinity of the generic and special fibres of \(F\). They are also of interest for the Jacobian conjecture since a result of S. Kaliman [Proc. Am. Math. Soc. 117, No. 1, 45-51 (1993; Zbl 0782.13017)] states that in order to prove the Jacobian conjecture (that is, for \(F,G \in \mathbb{C} [X,Y]\) if the Jacobian of \((F,G)\) is one, then \((F,G)\) is an automorphism of \(\mathbb{C}^ 2)\) it is enough to consider \(F\) and \(G\) with all fibers irreducible.
The construction gives for each positive integer \(n\) families \(F_{\mathbf a}\) \(({\mathbf a} \in \mathbb{C}^ m)\) of polynomials of degree \(6n + 4\) whose generic fibers have genus \(n\) and fixed singularities at infinite. Then by performing additional satellizations in the link at infinite of the special fibers \(F_{\mathbf a}\) one gets \(\mu (F_{\mathbf a}) = 0\) for some values of \({\mathbf a}\). This is done in a direct way by means of birational transformations of \(\mathbb{C}^ 2\) and it is also explained by means of the Eisenbud-Neumann diagrams of the link at infinity.

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14E05 Rational and birational maps
32A05 Power series, series of functions of several complex variables
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