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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.

##### MSC:
 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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##### References:
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