# The Graph of an Amoeba

The idea is to somehow flow the amoeba down to its underlying "graph" (an example of which is illustrated in the above figure). To achieve this, we modify our map $\varphi$ by taking logarithms with arbitrary bases

{\begin{aligned}\varphi _{t}:\ &(\mathbb {C} ^{*})^{2}\rightarrow \mathbb {R} ^{2}\\&(x,y)\mapsto (\log _{t}{|x|},\log _{t}{|y|})=\left({\dfrac {\log {|x|}}{\log {t}}},{\dfrac {\log {|y|}}{\log {t}}}\right)\end{aligned}} and then letting $t\rightarrow \infty$ . As $t$ increases this forces the central "belly" of the amoeba into a smaller and smaller region, whereas the thin "tentacles" take up a greater proportion of the space. In the limit we should get the graph of the amoeba.

However there is a snag: if the central "vertex" of our amoeba is not at the origin (consider for example the previous figure, where the vertex is at $(C-A,C-B)$ ) then this limiting process will end up moving the vertex to the origin, since all the scaling happens relative to the co-ordinate axes. To get around this we replace our curve $C$ by a family of curves $C_{t}$ . In the example of a generic line considered above, this family is given by:

$C_{t}=\{t^{A}x+t^{B}y+t^{C}=0\}\subseteq (\mathbb {C} ^{*})^{2}$ We then take the limit of the amoebas $\varphi _{t}(C_{t})$ as $t\rightarrow \infty$ and this gives us the graph as desired. This will be called the "'tropical curve"' associated to $(C_{t})$ , or the "'tropicalisation"' of $(C_{t})$ . We think of this as some sort of degeneration of the original curves.

Thus, the natural objects to tropicalise are not individual curves but rather families of curves $(C_{t})$ depending on a parameter $t\in \mathbb {R}$ . However, we cannot tropicalise every such family: we require that the equations for $C_{t}$ only involve (possibly fractional) powers of $t$ . (The reason for this will become clearer when we consider tropical curves in the context of Puiseux series: see Section Puiseux Series.)

Despite the definition being in terms of families, we will often want to tropicalise a single curve $C$ (as in the above example of a generic line). To do this, we proceed as we did in this example, by first passing to the naturally associated family $(C_{t})$ and then taking the tropicalisation of this family. We call the resulting tropical curve the "'tropicalisation"' of $C$ .

Computing tropicalisations using the current definition is no easy task. It would be good if we could find a more explicit expression for the tropicalisation of a curve $C$ , which doesn't require us to construct the family $(C_{t})$ . This leads us to the notion of a tropical polynomial and its corner locus.