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Amoebas of algebraic varieties

Definition and first properties

Take an algebraic variety $ Z$ in $ (\mathbb{C}\setminus{0})^n$. Its amoeba $ {\cal A}(Z)$ is its image by the map 

$\displaystyle {\rm Log}:(\mathbb{C}\setminus{0})^n$$\displaystyle \to$$\displaystyle \mathbb{R}^n$ 
$\displaystyle (z_1,\ldots,z_n)$$\displaystyle \mapsto$$\displaystyle (\log\vert z_1\vert,\ldots,\log\vert z_n\vert)$ 

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

$\displaystyle f=\sum_{{\bf\omega} \in I} b_{\bf\omega} {\bf z}^{\bf\omega}$
in $ \mathbb{C}[{\bf z},{\bf z}^{-1}]$, where the bold letters stand for $ n$-coordinate indeterminates (e.g $ {\bf z}=(z_1,\ldots,z_n)$); $ I$ is a finite subset of $ \mathbb{Z}^n$ and $ {\bf z}^{\bf\omega}$ means$ z_1^{\omega_1}\ldots z_n^{\omega_n}$.

Let $ Z_f$ be its zero set in $ (\mathbb{C}\setminus{0})^n$. We study its amoeba $ {\cal A}(Z_f)$. See an example of the picture of such an object in Figure 1.1

Figure 1.1: Example of an amoeba (taken from[Mik01])
\begin{figure}\epsfxsize =3.5cm \begin{center}\epsfbox{pictures/am1.eps}\end{center}\end{figure}

Connected components of the complement

Theorem 1.1   Any connected component $ {\cal F}$ of $ \mathbb{R}^n\setminus {\cal A}(Z_f)$ is convex.

This is proved in [GKZ94]: it is because $ {\rm Log}^{-1}({\cal F})$ is a domain of convergence of a certain Laurent series expansion of $ 1/f$.

A useful function is the Ronkin function for the hypersurface: it is the function $ N_f:\mathbb{R}^n\to \mathbb{R}$ defined by:

$\displaystyle N_f({\bf x})=\frac{1}{(2\pi i)^n}\int_{{\rm Log}^{-1}({\bf x})}\l... ... f({\bf z})\vert\frac{{\rm d}z_1}{z_1}\wedge\ldots\wedge\frac{{\rm d}z_n}{z_n}.$
Theorem 1.2 (Ronkin)   The Ronkin function is convex. It is affine on each connected component of $ \mathbb{R}^n\setminus {\cal A}(Z_f)$ and strictly convex on $ {\cal A}(Z_f)$.

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 $ \mathbb{R}^n\setminus {\cal A}(Z_f)$ after the following propositions.

Proposition 1.3   The derivative of $ N_f$ with respect to $ x_j$ is the real part of
$\displaystyle \nu_j({\bf x})= \frac{1}{(2\pi i)^n}\int_{{\rm Log}^{-1}({\bf x})... ...z_j}{f({\bf z})}\frac{{\rm d}z_1}{z_1}\wedge\ldots\wedge\frac{{\rm d}z_n}{z_n}.$
Proof. Write the coordinates in polar coordinates $ z_k=e^{x_k+i\theta_k}$. Then for fixed $ x_k$$ {\rm d} z_k/z_k=i{\rm d}\theta_k$ and we have
$\displaystyle (2\pi i)^nN_f({\bf x})=\int_0^{2\pi}\ldots\int_0^{2\pi}\log\vert ... ...ts,e^{x_n+i\theta_n})\vert i^n{\rm d}\theta_1\wedge\ldots\wedge{\rm d}\theta_n.$
Differentiating with respect to $ x_1$, we get 
$\displaystyle (2\pi i)^n\frac{\partial N_f}{\partial x_1}({\bf x})$$\displaystyle =$$\displaystyle \int_0^{2\pi}\ldots\int_0^{2\pi}Re\left(\frac{\partial f}{\partia... ...theta_1}}{f({\bf z})}\right)i^n{\rm d}\theta_1\wedge\ldots\wedge{\rm d}\theta_n$ 
 $\displaystyle =$$\displaystyle \int_{{\rm Log}^{-1}({\bf x})}Re\left(\frac{\partial f}{\partial ... ...{\bf z})}\right)\frac{{\rm d}z_1}{z_1}\wedge\ldots\wedge\frac{{\rm d}z_n}{z_n}.$ 

$ \qedsymbol$

For $ {\bf x}$ in a connected component $ {\cal F}$ of $ \mathbb{R}^n\setminus {\cal A}(Z_f)$, this is constant (since the homology class of the cycle $ {\rm Log}^{-1}({\bf x})$ in$ H_n\left((\mathbb{C}\setminus\{0\})^n\setminus V\right)$ remains unchanged) and was defined in [FPT00] to be the order of the component $ {\cal F}$. They proved the following properties, all based on the residue formula (other proofs in [Rul01]):

Proposition 1.4   For $ {\bf x}$ in a component $ {\cal F}$$ \nu_j$ is an integer.
Proof. Consider for fixed $ \theta_k$ ($ k\neq j$), the integral
$\displaystyle \frac{1}{2\pi i}\int_{\vert z_j\vert=e^{x_j}}\frac{\partial f}{\partial z_j}\frac{1}{f({\bf z})}{\rm d}z_j.$
By the residue formula this is an integer (it counts the number of zeroes of the function $ z_j\mapsto f(z_1,\ldots,z_n)$ minus the numbers of poles, in the disk of boundary $ \vert z_j\vert=e^{x_j}$), and since it depends continuously on the $ \theta_k$ ($ k\neq j$), it is independent of them.

It is equal to $ \nu_j$. Indeed,

$\displaystyle (2\pi i)^{n-1}\nu_1({\bf x})= \int_0^{2\pi}\ldots\int_0^{2\pi}\le... ...{f({\bf z})}{\rm d}z_1\right) {\rm d}\theta_2\wedge\ldots\wedge{\rm d}\theta_n.$
$ \qedsymbol$

Note that the fact that $ \nu_j$ is constant over any connected component of the complement implies that the partial derivatives of $ N_f$ in each such connected component are constant, hence $ N_f$ is affine there!

Proposition 1.5 (Proposition 2.4 in [FPT00])   $ \nu=(\nu_1({\bf x}),\ldots,\nu_n({\bf x}))$ is a lattice point of the Newton polygon $ \Delta$ of $ f$ (that is, the convex hull of the elements $ \omega$ of $ I$ for which $ b_\omega\neq 0$.)
Proof. The vector $ \nu$ is in $ \Delta$ if and only if for any vector $ {\bf s}\in\mathbb{Z}^n\setminus\{0\}$$ {\bf s}.\nu\leq\max_{\omega\in\Delta}{\bf s}.\omega$.

Indeed, $ \nu$ is in $ \Delta$ if and only if for any line $ l$ passing through 0, its orthogonal projection on $ l$ belongs to the projection of $ \Delta$ on $ l$ (see Figure 1.2). By density we can assume that $ l$ has a rational slope. The vector $ s$ appearing here represents the slope of $ l$, and the scalar product can be seen as the projection on $ l$.

Figure 1.2: Condition for $ \nu$ to belong to $ \Delta$
\begin{figure}\epsfxsize =3cm \begin{center}\epsfbox{pictures/projdelta.eps}\end{center}\end{figure}

Claim: $ {\bf s}.\nu$ is the number of zeroes (minus the order of the pole at the origin) of the one-variable Laurent polynomial $ w\mapsto f(({\bf c}w)^{\bf s})$ inside the unit circle $ \{\vert w\vert=1\}$ (where $ {\bf c}$ is any point of $ {\rm Log}^{-1}({\bf x})$$ {\bf x}$ being the point where $ \nu$ is computed).

But this polynomial has top degree equal to $ \max_{\omega\in\Delta}{\bf s}.\omega$. Hence we are done.

It remains to proof the claim. The numbers of zeroes (minus number of poles) of the function $ w\mapsto f((w{\bf c})^{\bf s})$ in the disk is given by the usual formula$ \frac{1}{2\pi i}\int_{\vert w\vert=1}\partial \log f((w{\bf c})^{\bf s})$. We use a change of variable formula $ w\mapsto(w{\bf c})^{\bf s}$. The image of the circle $ \{\vert w\vert=1\}$ by this change of variable is a loop in $ {\rm Log}^{-1}({\bf x})$, homologous to the sum $ s_1\gamma_1+\ldots+s_n\gamma_n$ where $ \gamma_j$ is the ``circle'' $ t\mapsto(c_1,\ldots,c_je^{2\pi t},\ldots,c_n)$ ($ t\in[0,1)$).


$\displaystyle \frac{1}{2\pi i}\int_{\vert w\vert=1}\partial \log f((w{\bf c})^{\bf s})$$\displaystyle =$$\displaystyle \sum_j s_j\int_{\gamma_j}\partial\log f(w)$ 
 $\displaystyle =$$\displaystyle \sum_j s_j\int_{\vert z_j\vert=e^{x_j}}\frac{\partial f}{\partial z_j}\frac{1}{f({\bf z})}{\rm d}z_j$ 
 $\displaystyle =$$\displaystyle 2\pi i\, {\bf s}.\nu\;.$ 

$ \qedsymbol$

Topologically it has the following meaning: $ {\rm Log}^{-1}({\bf x})$ is a $ n$-dimensional torus which does not intersect $ Z_f$. Consider for each $ j$ a loop $ \gamma_j$ of this torus (along which all the coordinates except $ z_j$ are constant), and let $ D_j$ be a disk whose boundary is $ \gamma_j$. Then $ \nu_j$ is the intersection number of $ Z_f$ and $ D_j$ (see also [Mik00]).

Theorem 1.6 (Proposition 2.5 in [FPT00])   The map 
$\displaystyle {\rm ind}: \mathbb{R}^n\setminus {\cal A}(Z_f)$$\displaystyle \to$$\displaystyle \Delta\cap\mathbb{Z}^n$ 
$\displaystyle {\bf x}$$\displaystyle \mapsto$$\displaystyle (\nu_1({\bf x}),\ldots,\nu_n({\bf x}))$ 

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 $ \Delta$.

Proof. Take two points $ {\bf x}$ and $ {\bf x'}$ in $ \mathbb{Q}^n\setminus {\cal A}(Z_f)$, and let $ \nu={\rm ind}({\bf x})$ and $ \nu'={\rm ind}({\bf x'})$. Let $ {\bf s}\in\mathbb{Z}^n\setminus\{0\}$ such that $ {\bf x'}={\bf x}+r{\bf s}$ for some positive $ r$. The claim in the preceding proof implies that $ {\bf s}.\nu$ and $ {\bf s}.\nu'$ are the numbers of zeroes inside $ \{\vert w\vert=1\}$ of the two polynomials$ w\mapsto f((w{\bf c})^{\bf s})$ and $ w\mapsto f((w{\bf c'})^{\bf s})$, where $ {\rm Log}({\bf c})={\bf x}$ and $ {\rm Log}({\bf c'})={\bf x'}$; we choose $ {\bf c'}$ such that $ c'_j/c_j=e^{rs_j}$ i.e. they have the same argument. Hence $ (w{\bf c'})^{\bf s}=(e^rw{\bf c})^{\bf s}$. Thus $ {\bf s}.\nu'$ is the number of zeroes of $ w\mapsto f((w{\bf c})^{\bf s})$ inside the circle $ \{\vert w\vert=e^r\}$.

If $ \nu=\nu'$, this means that $ w\mapsto f((w{\bf c})^{\bf s})$ has no zero in the ring $ R=\{1<\vert w\vert<e^r\}$, hence there is no point of the amoeba on the segment$ [{\bf x},{\bf x'}]$ (see Figure 1.3). This implies that $ {\bf x}$ and $ {\bf x'}$ are in the same component. $ \qedsymbol$

Figure 1.3: The image of the ring $ R$by the map $ w\mapsto {\bf c}w$ is sent to the segment $ [{\bf x,x'}]$
\begin{figure}\epsfxsize =5cm \begin{center}\epsfbox{pictures/couronne.eps}\end{center}\end{figure}


Define $ N_f^\infty=\max_{\cal F}N_{\cal F}$ where the $ {\cal F}$ range through the connected components of $ \mathbb{R}^n\setminus {\cal A}(Z_f)$ and $ N_{\cal F}:\mathbb{R}^n\to\mathbb{R}$ is the affine function whose restriction to $ {\cal F}$ coincides with $ N_f$.

Definition 1.7 ([PR00])   The spine of $ {\cal A}(Z_f)$ is the corner locus of the function $ N_f^\infty$.

As we will see later, this is a tropical variety.

It is a deformation retract of the amoeba $ {\cal A}(Z_f)$ (see [Rul01]). See Figure 1.4.

Figure 1.4: Amoeba and its spine (taken from[Mik01])
\begin{figure}\epsfxsize =3.5cm \begin{center}\epsfbox{pictures/am-spine.eps}\end{center}\end{figure}

Remark: In [PR00] and [Rul01], the (non-obvious) relation between the coefficients of $ f$ and the coefficients of the ``tropical polynomial'' $ N_f^\infty$ is studied (see later for the meaning of ``tropical polynomial'').

It is shown there that $ N_f^\infty({\bf x})=\max_{\omega\in J}\{{\rm Re}\,\Phi_\omega(f)+\omega.{\bf x}\}$ where $ J$ is the subset of $ I$ of the $ \omega$ for which there exist a connected component $ {\cal F}_\omega$ of order $ \omega$ of $ \mathbb{R}^n\setminus {\cal A}(Z_f)$, and $ \Phi_\omega(f)=\frac{1}{(2\pi i)^n}\int_{{\rm Log}^{-1}({\bf x})}\log[f({\bf z})/z^\omega]\frac{{\rm d} z_1}{z_1}\wedge\ldots\wedge\frac{{\rm d}z_n}{z_n}$ for $ {\bf x}\in {\cal F}_\omega$.

It is also proved that, in the particular case where $ I$ has no more than $ 2n$ points and that no $ k+2$ of these lie in an affine $ k$-dimensional subspace for $ k=1,\ldots,n-1$$ N_f^\infty({\bf x})=\max_{\omega\in I}\{\log\vert b_\omega\vert+\omega.{\bf x}\}$ (remember that the $ b_\omega$ are the coefficient of $ f$).

Monge-Ampère measure

see [PR00], [Rul01]...

Compactified Amoeba

Given the Newton polytope $ \Delta$, denote by $ A\subset I$ the set of vertices of $ \Delta$, and consider the ``moment map'': 

$\displaystyle \mu: (\mathbb{C}\setminus\{0\})^n$$\displaystyle \to$$\displaystyle \Delta$ 
$\displaystyle {\bf z}$$\displaystyle \mapsto$$\displaystyle \frac{\sum_{\omega\in A}\vert{\bf z}^\omega\vert\,\omega}{\sum_{\omega\in A}{\bf z}^\omega\vert}.$ 

In fact $ \mu$ is the restriction of the moment map $ \overline{\mu}: T_\Delta\to\Delta$ where $ T_\Delta$ is the toric variety associated to $ \Delta$.

The compactified amoeba $ \overline{\cal A}(Z_f)$ of $ Z_f$ is the closure of $ \mu(Z_f)$ in $ \Delta$.

See [Mik01].

Figure 1.5: Compactified amoeba of the amoeba of figure 1.1 (taken from[Mik01])
\begin{figure}\epsfxsize =3.5cm \begin{center}\epsfbox{pictures/am2.eps}\end{center}\end{figure}

First application: Harnack curves

Definition 1.8   A curve $ \mathbb{R}X$ of degree $ d$ in $ \mathbb{R}P^2$ is in maximal position with respect to the (generic) lines $ l_1,l_2,l_3$ if 
$ \bullet$ $ \mathbb{R}X$ is maximal (maximal number of ovals) 
$ \bullet$ There exist three disjoints arcs $ r_1,r_2,r_3$ on one connected component such that $ \char93 (r_i\cap l_i)=d$.
Theorem 1.9 (Mikhalkin)   $ \forall d>0$, there exist only one maximal topological type (Harnack curve). If the number of generic lines is greater than $ 3$, there is no such maximal topological type.

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

Figure 1.6: The maximal topological type for $ d=10$(taken from[Mik01])
\begin{figure}\epsfxsize =5cm \begin{center}\epsfbox{pictures/deg10.eps}\end{center}\end{figure}

Second application: dimers

see [KO],[KOS].

Benoit BERTRAND 2003-12-19