Subsections

Amoebas of algebraic varieties

Definition and first properties

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

in , where the bold letters stand for -coordinate indeterminates (e.g ); is a finite subset of and means .

Let be its zero set in . We study its amoeba . See an example of the picture of such an object in Figure 1.1

Connected components of the complement

Theorem 1.1   Any connected component of is convex.

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:

Theorem 1.2 (Ronkin)   The Ronkin function is convex. It is affine on each connected component of and strictly convex on .

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.

Proposition 1.3   The derivative of with respect to is the real part of

Proof. Write the coordinates in polar coordinates . Then for fixed , and we have

Differentiating with respect to , we get

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]):

Proposition 1.4   For in a component , is an integer.

Proof. Consider for fixed (), the integral

By the residue formula this is an integer (it counts the number of zeroes of the function minus the numbers of poles, in the disk of boundary ), and since it depends continuously on the (), it is independent of them.

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!

Proposition 1.5 (Proposition 2.4 in [FPT00])   is a lattice point of the Newton polygon of (that is, the convex hull of the elements of for which .)

Proof. The vector is in if and only if for any vector , .

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'' ().

Hence

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]).

Theorem 1.6 (Proposition 2.5 in [FPT00])   The map

sends two different connected components to two different points.

This implies that the number of connected components is finite, and less than or equal to the number of lattice points in .

Proof. Take two points and in , and let and . Let such that for some positive . The claim in the preceding proof implies that and are the numbers of zeroes inside of the two polynomials and , where and ; we choose such that i.e. they have the same argument. Hence . Thus is the number of zeroes of inside the circle .

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.

Spine

Define where the range through the connected components of and is the affine function whose restriction to coincides with .

Definition 1.7 ([PR00])   The spine of is the corner locus of the function .

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 ).

Monge-Ampère measure

see [PR00], [Rul01]...

Compactified Amoeba

Given the Newton polytope , denote by the set of vertices of , and consider the moment map'':

In fact is the restriction of the moment map where is the toric variety associated to .

The compactified amoeba of is the closure of in .

See [Mik01].

First application: Harnack curves

Definition 1.8   A curve of degree in is in maximal position with respect to the (generic) lines if
is maximal (maximal number of ovals)
There exist three disjoints arcs on one connected component such that .

Theorem 1.9 (Mikhalkin)   , there exist only one maximal topological type (Harnack curve). If the number of generic lines is greater than , there is no such maximal topological type.

see [Ite03], [Mik00], [Mik01].

Second application: dimers

see [KO],[KOS].

Benoit BERTRAND 2003-12-19