Integrable Hamiltonian systems with incomplete flows and Newton's polygons

TL;DR

This paper investigates Hamiltonian vector fields defined by polynomials in two complex variables, focusing on their behavior at infinity using Newton's polygons, and constructs a compactification to analyze singularities.

Contribution

It introduces a coordinate system near infinity for such Hamiltonian systems and classifies singularities based on Newton's polygons, providing a new geometric framework.

Findings

Canonical form of $f(z,w)$ near infinity

Construction of a compactification of the level set

Classification of singularities via Newton's polygons

Abstract

We study the Hamiltonian vector field $v=(-\partial f/\partial w,\partial f/\partial z)$ on $\mathbb C^2$, where $f=f(z,w)$ is a polynomial in two complex variables, which is non-degenerate with respect to its Newton's polygon. We introduce coordinates in four-dimensional neighbourhoods of the "points at infinity", in which the function $f(z,w)$ and the 2-form $dz\wedge dw$ have a canonical form. A compactification of a four-dimensional neighbourhood of the non-singular level set $T_0=f^{-1}(0)$ of $f$ is constructed. The singularity types of the vector field $v|_{T_0}$ at the "points at infinity" in terms of Newton's polygon are determined.