Integrable Hamiltonian systems with incomplete flows and Newton's polygons

Details Integrable Hamiltonian systems with incomplete flows and Newton's polygons Integrable Hamiltonian systems with incomplete flows and Newton's polygons

2011-07-11

arxiv.org/abs/1107.1911v2

Contemp. Probl. Math. Mech. VI (2011), N.3, 42-55, Moscow Univ. Publ., 2011

Mathematics

Differential Geometry

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

Scientific paper

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.

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