Let $f$ be a $d$-variable complex Laurent polynomial; consider
$\{z \in (C^*)^d|f(z)=0\}$.  Consider the map $\Log:(C^*)^d \to \R^d$,
$z \mapsto (\log |z_1|,\dots,\log |z_d|)$.

Def'n: $\Log(f=0) \subset \R^d$ is the {\em amoeba} of $f$.

Remark:

Let $A \subset \Z^d$ be the finite set of nonzero monomials in $f$; let
$P$ be the convex hull of $A$, and consider the projective toric variety
$Y_P$ with line bundle $\L_P$.   There is a section $\hat f$ of $\L_P$ such
that $\hat f^{-1}(0)$ is a compactification of $f^{-1}(0)$.

We have the following diagram:

(C^*)^d \to X_P
  |         |
  |Log      | moment map $\mu$
  v         v
\R^d   \to  P

In particular, $\Log(\{f=0\}) \subset \R^d$ is isomorphic to
$\Int(P)$.

Ex: Let $f(x,y)=x+y-1$.  The amoeba is the image of 
$\C-\{0,1\} \to \R^2$ sending $y$ to $(\log |1-y|,\log |y|)$.

Critical points: $y \in \R-\{0,1\}$.

The range is the "interior" of the image of $\R-\{0,1\}$ (goes to $\infty$
at 3 points)..

What do we generally know about amoebas?

Lemma: Let $O=\Log(\{f=0\})$.  Each connected component of $\R^d-O$ is
convex.  Moreover, there is a bijective correspondence between components
and Laurent expansions of $1/f$.

Pf: 

"Recall" the following facts from complex analysis:
1) The convergence domain of any $d$-variable Laurent series is of the form
$\Log^{-1}(U)$, where $U$ is convex.
2) Let $U \subset \R^n$ be convex, $g:\Log^{-1}(U) \to \C$ holomorphic. Then
$g$ has a convergent Laurent expansion on $U$.

(We can think of these as generalizations of the statements that Laurent
expansions converge on annuli, and functions on annuli have Laurent series
that converge at least there).

Apply these facts to $1/f$.
$1/f$ is holomorphic on $\Log^{-1}$ of the
complement of the amoeba...

"I'm really mystified; sorry."

Exercise!

Now, let $f=\sum_{w \in A} a_w z_1^{w_1} \dots z_d^{w_d}$, $a_w \neq 0$;
take a vertex of the corresponding convex polytope $P$.  Then we can write

$f(z)=a_v z^v \sum_{w \in A} a_w/a_v z^{w-v}$; so we can write

$1/f=(a_v z^v)^{-1}(1+\sum_{w \neq v} a_w/a_v z^{w-v})^{-1}$, expanded
as a geometric series; as $v$ is
a vertex, all the other elements of $A$ lie in some half-plane, and
so higher powers of them will lie deeper in the half-plane (so, in particular,
any given monomial can only appear finitely many times).

Let $N_v$ be the normal cone of $v$.  Then the above expansion converges
in $\Log^{-1}(b_v+N_v)$ for some $b_v \in N_v$.

Pf: Let $g(z)=\sum_{w \neq v} a_w/a_v z^{w-v}$.  Then the expansion converges
if $|g(z)|<1$.  Recall that $N_v=\{\lambda|<\lambda, v-w> \geq 0
for all w \in P\}$

Assume $z \in \Log^{-1}(N_v)$, and consider $|z|^{w-v}$.
Let $\Log(z)=\lambda$; then $|z|^{w-v}=e^{<\lambda,w-v>} \leq 0$;
indeed, by shifting by some $b_v$, we can make it as small as necessary.
So apply the triangle inequality to $g$...

Consequence: The amoeba $\Log(\{f=0\})$ is contained in
$\bigcup_{v a vertex} (b_0+N_v)$.

What is the relationship between non-archimedean and archimedean geometry?

Thm (Maslov's dequantization): 
Recall: Last time we had $K=\C[[t]](t^{-1},t^{1/k})$, with a valuation
$\val \mapsto \Q \cup {-\infty}$.

Additionally, we have the map $\C[t] \mapsto \K$, and the evaluation
maps $s_\epsilon:\C[t] \to \C$.  Define $l_\epsilon(c)=\log |c|/\log \epsilon$.

We have the diagram

\C<--------------\C[t]------------->\K
|                                   |
|                                   |
v                                   v
\R \cup \{\infty\}<----------\Q \cup \{\infty\}

where the vertical maps are $-\val$ and $l_\epsilon$, and the bottom map is
inclusion.

Clearly, this diagram doesn't commute.  However, $\lim_{\epsilon \to 0}
l_\epsilon s_\epsilon(f)=-\val(f)$ (elementary analysis exercise).  So it
"commutes in the limit".

The corresponding multivariable statement:

Take $f$ a Laurent polynomial, with some additional variable $t$.

We can either think of $f$ as a Laurent polynomial over $\K$, in which case
it has a non-archimedean amoeba $\Omega$, 
or we can take $f_{\epsilon}=f(\epsilon,
dots)$, and associate to it the archimedean amoeba $O_\epsilon$, defined
using $\Log_\epsilon=\Log/\log |\epsilon$.

Then we have

Them (Rullgard-Mikhalkin):

$\lim_{\epsilon \to 0} O_\epsilon=\Omega$ (in the compactly supported
Hausdorff topology on $\R^d$).

What does this mean?

If we want to construct an amoeba that looks like something specific, we
can build a non-archimedean amoeba that looks like that (this is a finite
combinatorial process) and then set $t=\epsilon$ for some sufficiently small
$\epsilon$.

Ex: Let $P$ be the polytope with vertices $(0,0)$, $(0,d)$, $(d,0)$;
let $A=P \cap \Z^2$.  If $f$ has support $A$, $f^{-1}(0) \subset \C P^2$
is a curve of degree $d$.  Choose a $\nu:A \to \Z$ (giving
exponents $t^?$ in $K[stuff...]$) such that the resulting subdivision
$P=P_1 \cup \dots P_l$ is into minimal triangles with lattice-point
vertices.

This is what the weird exponents from the first lecture are for...

Then the nonarchimedean amoeba $\Omega$ is "dual" to this subdivision, and
$O_\epsilon$ is a fattened version of $\Omega$.  (This is more information
than Hausdorff convergence gives you; to see that it holds, can make an 
argument that there can't be more holes in the amoeba than integer
points in the polytope).

Another perspective:

We start with a finite set $A \subset \Z^d$, and a weight function
$\nu:A \to \Z$ (note that it's integral now).

We have the Newton polytope $P$, the extended Newton polytope $\Gamma
\subset \R \times \R^d$, and $P=P_1 \cup \dots \cup P_k$
given by $v$.

Question: We know that, given $P$, can obtain a projective toric variety.

What do we associate to $\Gamma$?

Define a graded ring $R$ by $R^k=\C[k \Gamma \cap
\Z^{d+1}]=\C[\Gamma \cap \frac{1}{k} \Z^{d+1}]$, except $R^0=\C[N_0
\times (0,0,\dots)^d]$.

From $R=R_\Gamma$ we get a variety $X=X_{\Gamma}$, a line bundle
$\L_\Gamma$ over $X$.  However, it's no longer a projective variety.

We have $\C[t] \to R_0 \subset R$; thus, dually, there is a map
$\pi: X_\Gamma \to \C=\A^1$.

What are the fibers of $\pi$?

General fiber: $\pi^{-1}(\A^1-\{0})=X_\Gamma^{gen}$.

On the level of homogenous coordinate rings, $R^{gen}_\Gamma=R_{\Gamma}
\otimes \C[t,t^{-1}]$.  Since multiplying by $t$ corresponds to moving up,
$t^{-1}$ corresponds to moving down, and so this is isomorphic to
$R_P \otimes_\C \C[t,t^{-1}]$; thus
$\pi^{-1}(A-\{0\})=X_P \times (\A^1-\{0\})$, and so each nonzero fiber
is $X_P$.

The zero fiber might also be $X_P$ (i.e., if the function is zero); on the
other hand, it also might not be...