Structure of the Newton tree at infinity of a polynomial in two variables

Let \(f:\mathbb{C}^2 \to \mathbb{C}\) be a polynomial map. Let \(\mathbb{C}^2 \subset X\) be a compactification of \(\mathbb{C}^2\) where \(X\) is a smooth rational compact surface and such that there exists a morphism of varieties \(\Phi :X\to \mathbb{P}^1\) which extends \(f\). Put \(\mathcal{D}=X...

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Bibliographic Details
Published in:arXiv.org 2019-04
Main Authors: Cassou-Nogues, Pierrette, Daigle, Daniel
Format: Article
Language:English
Subjects:
Polynomials
Singularities
Online Access:Get full text
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Summary:Let \(f:\mathbb{C}^2 \to \mathbb{C}\) be a polynomial map. Let \(\mathbb{C}^2 \subset X\) be a compactification of \(\mathbb{C}^2\) where \(X\) is a smooth rational compact surface and such that there exists a morphism of varieties \(\Phi :X\to \mathbb{P}^1\) which extends \(f\). Put \(\mathcal{D}=X\setminus \mathbb{C}^2\); \(\mathcal{D}\) is a curve whose irreducible components are smooth rational compact curves and all its singularities are ordinary double points. The dual graph of \(\mathcal{D}\) is a tree. We are interested in this tree, and we analyse its complexity in terms of the genus of the generic fiber of \(f\).
ISSN:2331-8422