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### Monge-Ampère

http://www.dm.unipi.it/~bertrand/amoeb-geotrop/node1.html
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# 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