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

the zero locus of a tropical polynomial

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The next question is to understand the zero set of a tropical function. When \(f\) is a polynomial, we have the associated variety \(V(f):= \{q\st f(q)=0\}\). Naïvely, 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 0.0.1

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. This mimics the following behavior. A tropical polynomial \(\phi: \RR^n\to \RR\) by definition, is a convex piecewise integral affine function on \(\RR^2\). We can associate to every such function the locus where \(\phi\) where \(\phi\) is not differentiable. This will be a union of polyhedra of dimension \(n-1\).

definition 0.0.2

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 \(q\)}\}.\] We now give some more details to the previous section.

definition 0.0.3

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 0.0.4

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 0.0.5

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.

definition 0.0.6

A set of polyhedra is called a polyhedral complex if
figure 0.0.7:Two examples of unions of polyhedra. On the left, a polyhedral complex. The right figure is not an example of a polyhedral complex

lemma 0.0.8

Every set of polyhedra has a polyhedral complex subdivision. Later, we will impose some additional criteria on these complexes:

definition 0.0.9

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