#
COMPUTATION AND APPLICATIONS OF THE NEWTON POLYHEDRON

**Alexander B. Aranson
**

*Department of Mathematics, Keldysh Institute of Applied Mathematics
*

*Miusskaja Square 4, Moscow 125047, Russia
*

gtrtp@glasnet.ru, aranson@spp.keldysh.ru

**Abstract**
We consider the multidimensional polynomial

where
,
or
,
,
,
is a *monomial*,
coeficients
or
and *D* is some set
in
.
The set
is called the *support* of the
polynomial *f*(*X*). The convex hull *M*=*M*(*f*) of the set *D* is called
the *Newton polyhedron* of the polynomial *f*(*X*).
There is some correspondence between properties of the polynomial *f*(*X*)
and of its Newton polyhedron *M*(*f*).
It was studied by Bruno, Soleev, Khovanskiy and somes other.
We propose algorithms and the computer programm for computation of the
Newton polyhedron of any multidimensional polynomial, and for computation
of all elements of this polyhedron (surfaces, vertices, edges, etc.).
We consider the application of the Newton polyhedron method for analysis
of the behavior of a 4-dimensional reversible ODE system near its equilibrium
point. This system appeared from Hydrodynamics after reduction on the center
manifold of the water-wade problem.

IMACS ACA'98 Electronic Proceedings