Arnold's seminar,
Decemvber 22, 1998

Vladlen Timorin.
Amoebas and Laurent series.

Summary of the talk.

This talk is based on the Doctoral Thesis of M. Forsberg and on 
a recent paper of G. Mikhalkin.

Let p be a polynomial in n complex variables. Consider the following 
map Ln: z ----> ln|z|. The image of the zero set {p=0} under this map 
is called the amoeba of the polynomial p. It turns out that the complement
to the amoeba splits into a finite number of convex components. 
This result and the definition of amoeba are due to Gelfand, Kapranov and 
Zelevinsky. 
The number of the components is bounded from above by the number of 
integer points in the Newton polytope N. Each vertex of p
corresponds to a component. There may be several 
components that correspond to interior integer points of N or to relative 
interiors of some faces of N. But existence of such components depends 
on the coefficients of p.
The following results will presumably be explained:
1) The components of the complement to the amoeba coincide with 
the domains of convergence of Laurent expansions for 1/p.
2) There is a universal amoeba constructed by the polynomial p
with additional variables instead of coefficients. Its complement
has the same number of components as p has monomials.
3) A 2-dimensional amoeba looks like the dual graph to 
a certain subdivision of the Newton polytope. 
4) The degree of the logarithmic Gauss map from {p=0} to CP^n
equals to n!Vol(N).