On polynomials whose fibers are irreducible with no critical points
type: Item
instance of: scholarly article
title: On polynomials whose fibers are irreducible with no critical points
zbMATH Open document ID: 0803.13009
DOI: 10.1007/BF01459795
author: Pi. Cassou-Noguès
author: Ignacio Luengo
author: Enrique Artal Bartolo
published in: Mathematische Annalen
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165219
review text: 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 \textit{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.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 32A05
zbMATH DE Number: 683110
zbMATH Keywords: automorphism of complex plane
zbMATH Keywords: polynomial mapping
zbMATH Keywords: Jacobian conjecture
reviewed by: Enrique Artal Bartolo
MaRDI profile type: Publication
cites work: Entrelacs toriques itérés et intégrales associées à une courbe plane
cites work: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. (AM-110)
cites work: On the Jacobian Conjecture
cites work: Complex algebraic plane curves via their links at infinity
cites work: On the boundary obstructions to the Jacobian problem
cites work: Sur les opérations holomorphes du groupe additif complexe sur l'espace de deux variables complexes
cites work: The number of reducible hypersurfaces in a pencil. -- Erratum
OpenAlex ID: W1527248963
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description: scientific article; zbMATH DE number 683110