\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: a short introduction to tropical geometry

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 0.0.1

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 \(q_1, q_2\in \TT\).
The goal is now to understand how algebraic geometry over \(\CC\) relates to ``algebraic geometry'' over the semi-field \(\TT\). The first thing to do it to exchange polynomials for tropical polynomials. For instance, given \(f(z_1, z_2): \CC^2\to \CC\) a polynomial of two variables, we declare the tropicalization of the polynomial to be \[f(z_1, z_2)=\sum a_{ij} z_1^iz_2^j \leftrightarrow \TropB(f)(q_1, q_2):= \max (a_{ij}+iq_1 + jq_2 )\]

example 0.0.2

If \(f=q_1+q_2+1\), the tropicalization is given by \(\TropB(f) = \max(1+q_1, 1+q_2, 1)\)