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

Abstract

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 $f$.

Changes from v1: we added section 7 and rewrote the introduction