### 1: a short introduction to tropical geometry

We first study the log map \[ \Log_t:= \log_t|-|: \CC\to \RR\cup\{-\infty\} \] where \(t>1\). and try to define a ``ring structure'' on \(\RR\cup\{-\infty\}\) which makes this log map a homomorphism. The first guess that one would take is to define \begin{align*} ``\times" \text{ given by the operation } q_1 ``\times" q_2 = &\Log_t(\Log_t^{-1}(q_1) \cdot \Log_t^{-1}(q_2)) =q_1+q_2 \\ ``+" \text{ given by the operation } q_1 ``+" q_2 = &\Log_t(\Log_t^{-1}(q_1) + \Log_t^{-1}(q_2)) \end{align*} While the first operation is well defined, the second is not! In order to make this well defined we take the limit of the second equation as \(t\to \infty\), from which we obtain \[\lim_{t\to\infty} \Log_t(\Log_t^{-1}(q_1) + \Log_t^{-1}(q_2)) = \max(q_1, q_2).\] This gives us a definition for our new operations.## definition 1.0.1

The*semi-field of tropical numbers*is the set \(\TT:= \RR\cup_\infty\) equipped with the operations (called tropical plus and tropical times): \begin{align*} q_1\oplus q_2 =& \max(q_1, q_2)\\ q_1\odot q_2 =& q_1+q_2 \end{align*} where \(q_1, q_2\in \TT\).

## example 1.0.2

If \(f=q_1+q_2+1\), the tropicalization is given by \(\TropB(f) = \max(1+q_1, 1+q_2, 1)\)### 2: the zero locus of a tropical polynomial

The next question is to understand the zero set of a tropical function. When \(f\) is a polynomial, we have the associated variety \(V(f):= \{q\st f(q)=0\}\). Na\"ively, one might try define the tropical zero set to be the set where \(\TropB(f)=-\infty\) (as \(-\infty\) is the tropical additive identity). However, it is clear to see that this will have no solutions.## example 2.0.1

Take \(f=q_1+q_2+1\), where \(q_1,q_2 \in (\CC^*)^2\). Drawing the image of \(V(f)\) under \(\Log_t\) gives us the following sequence of images: We call the image of \(V(f)\) under the \(\Log_t\) map the*amoeba*of \(V(f)\), which (in this example) appears to converge to some piecewise linear object.

## definition 2.0.2

Let \(\phi: \RR^n\to \RR\) be a tropical polynomial. The*tropical zero set*is set \[V(f):=\{q\in \RR^n \text{ such that \(f\) is not differentiable at \(q\)}\}.\]

## definition 2.0.3

Fix a lattice structure \(\ZZ^n\subset \RR^n\) and \(\ZZ\subset \RR\). We say that a function \(\underline \phi: \RR^n\to \RR\) is*integral affine*if \(\underline \phi= \phi_{l}+c\) where \(\phi_{l}\) is \(\ZZ\)-linear and \(c\in \RR\) is some constant.

## definition 2.0.4

For \(\underline \phi: \RR^n\to \RR\) a integral affine structure, define \(H_{\underline \phi}:= \{q\in \RR^n \st \underline\phi(q) \geq 0 \}.\)## definition 2.0.5

We say that \(\sigma\subset \RR^n\) is a*rational polyhedron*if \(\sigma = \bigcap_{i\in I} H_{\underline \phi_i}\) for some collection of integral affine functions.

- We say that it is a cone if a translate of it is closed under multiplication by \(\RR_{\geq 0}\)
- We say that \(\sigma\) is a polytope if it is compact.

## definition 2.0.6

A set of polyhedra is called a*polyhedral complex*if

- if \(\sigma\in P\), then every face \(\tau< \sigma\) is also contained in \(P\),
- For \(\sigma_1, \sigma_2\in P\): if \(\sigma_1\cap\sigma_2= \tau \neq \emptyset\),then \(\tau\) is a face of both \(\sigma\) and \(\sigma'\).

## lemma 2.0.8

Every set of polyhedra has a polyhedral complex subdivision.- Pure dimension \(k\) --- every maximal cell is of dimension \(k\),
- ``Weighted'': there exists a weight function \(\weight: \{\text{Maximal Cells}\} \to \ZZ\)
- ``Balanced'': for a weighted pure complex: every codimension 1 face \(\tau\) is we have \(\sum_{\sigma} \weight(\sigma) v_{\sigma/\tau}=0\) in \(\RR^n /\Span(\tau)\).

## definition 2.0.9

A*tropical \(k\)-cycle*in \(\RR^n\) is a balanced weighted polyhedral complex.

### 3: a sketch of intersection theory

Given \(Z_k(\RR^n)\), the set of tropical \(k\)-cycles, has a group structure, whose additive structure is given by taking unions and subdividing. This also comes with a ring structure \begin{align*} Z_k(\RR^n)\otimes Z_l(\RR^n)\to \ZZ_{n+l-n}(\RR^n)\\ X\tensor Y \mapsto \lim_{\epsilon\to 0} (X\cap (T+\eps v)) \end{align*} where is chose so that the intersection is transverse for all \(\epsilon>0\) sufficiently small. The key step to proving that this ring structure is well defined is the moving lemma.## lemma 3.0.2

The content of the moving lemma goes here## definition 3.0.3

A function \(\psi: \RR^n\to \RR\) is a*tropical rational function*is a piecewise integral affine function.

## example 3.0.4

The function \(\psi(q_1, q_2):= \text{min}(1+q_1, 1+q_2, 1)\) is an example of a rational function. Observe that this is not a convex function, and therefore is*not*a tropical polynomial

## definition 3.0.5

\begin{align*} R_k:= &\{\div(\psi)_S\cdot c : \psi\in PA(\RR^n), c\in Z_{k+1}(\RR^n)\}\\ R_k^b:=&\{\div(\psi)_S\cdot c : \psi\in PA(\RR^n), \text{bounded} , c\in Z_{k+1}(\RR^n)\} \end{align*}### 4: a sketch of compactification in tropical geometry

Note that we have some ideas of compactification already: after all we can define tropical polynomials on \(\RR^n\), and \(\TT^n\) is a compactification of \(\RR^n\). The space \(\TT^n\) has a decomposition into pieces \(\RR_I\), which we call the torus orbits. The key notion that we will need to study is how many ``-'infinity'''s does a point \(x\) belong to.## definition 4.0.1

Given \(x\in \TT^n\), we define the*sedentarity*of \(x\) to be the number of \(-\infty\)'s belonging to \(q\) in its coordinate description.