Integrable Hamiltonian systems with incomplete flows and 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.

14 pages, 6 figures, in Russian, Proceedings of International Conference "Metric geometry of surfaces and polytopes" (Moscow, Aug. 2010)

References in corpus (1)

• Connected components of the moduli spaces of Abelian differentials with prescribed singularities