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 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.
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 0.0.6
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 \(\sigma'\).
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:
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 0.0.9
A tropical \(k\)-cycle in \(\RR^n\) is a balanced weighted polyhedral complex.
