Take an algebraic variety in
. Its amoeba
is its image by the map
This name was first introduced by Gelfand, Kapranov and Zelevinsky in [GKZ94].
A first property of an amoeba is that it is closed.
Most of the properties we will mention concern amoebas of hypersurfaces, so that from now on, unless otherwise specified, we will consider a Laurent polynomial
Let be its zero set in . We study its amoeba . See an example of the picture of such an object in Figure 1.1
This is proved in [GKZ94]: it is because is a domain of convergence of a certain Laurent series expansion of .
A useful function is the Ronkin function for the hypersurface: it is the function defined by:
See [PR00] for the study of the Ronkin function.
Actually, we will be able to see that it is affine on each connected component of after the following propositions.
For in a connected component of , this is constant (since the homology class of the cycle in remains unchanged) and was defined in [FPT00] to be the order of the component . They proved the following properties, all based on the residue formula (other proofs in [Rul01]):
It is equal to . Indeed,
Note that the fact that is constant over any connected component of the complement implies that the partial derivatives of in each such connected component are constant, hence is affine there!
Indeed, is in if and only if for any line passing through 0, its orthogonal projection on belongs to the projection of on (see Figure 1.2). By density we can assume that has a rational slope. The vector appearing here represents the slope of , and the scalar product can be seen as the projection on .
Claim: is the number of zeroes (minus the order of the pole at the origin) of the one-variable Laurent polynomial inside the unit circle (where is any point of , being the point where is computed).
But this polynomial has top degree equal to . Hence we are done.
It remains to proof the claim. The numbers of zeroes (minus number of poles) of the function in the disk is given by the usual formula . We use a change of variable formula . The image of the circle by this change of variable is a loop in , homologous to the sum where is the ``circle'' ().
Topologically it has the following meaning: is a -dimensional torus which does not intersect . Consider for each a loop of this torus (along which all the coordinates except are constant), and let be a disk whose boundary is . Then is the intersection number of and (see also [Mik00]).
This implies that the number of connected components is finite, and less than or equal to the number of lattice points in .
If , this means that has no zero in the ring , hence there is no point of the amoeba on the segment (see Figure 1.3). This implies that and are in the same component.
Define where the range through the connected components of and is the affine function whose restriction to coincides with .
As we will see later, this is a tropical variety.
It is a deformation retract of the amoeba (see [Rul01]). See Figure 1.4.
Remark: In [PR00] and [Rul01], the (non-obvious) relation between the coefficients of and the coefficients of the ``tropical polynomial'' is studied (see later for the meaning of ``tropical polynomial'').
It is shown there that where is the subset of of the for which there exist a connected component of order of , and for .
It is also proved that, in the particular case where has no more than points and that no of these lie in an affine -dimensional subspace for , (remember that the are the coefficient of ).
see [PR00], [Rul01]...
Given the Newton polytope , denote by
the set of
vertices of , and consider the ``moment map'':
The compactified amoeba of is the closure of in .
see [Ite03], [Mik00], [Mik01].
Benoit BERTRAND 2003-12-19