# Puiseux Series

In Section The Graph of an Amoeba we introduced in a rather ad hoc way families of curves $(C_{t})$ , in order to avoid all the vertices of the tropical curve getting sent to the origin. There is a more elegant way of constructing tropicalisations in which the appearance of the family $(C_{t})$ is far more natural: namely, we think of this family of curves in $(\mathbb {C} ^{*})^{2}$ as being a single curve in $K^{2}$ , where $K$ is the field of Puiseux series.

The Puiseux series are defined to be formal power series (with coefficients in $\mathbb {C}$ ) indexed by $\mathbb {Q}$ $\sum _{q\in \mathbb {Q} }a_{q}t^{q}$ such that the set of $q\in \mathbb {Q}$ with $a_{q}\neq 0$ is bounded above and has only a finite number of denominators appearing in its elements. This is an algebraically closed field of characteristic zero, and so when we do algebraic geometry over $K$ the resulting theory will be similar to that of the complex numbers.

Coming back to the family of curves $(C_{t})$ , we now view the parameter $t$ appearing in the equation for $C_{t}$ as the formal variable in $K$ . Thus the family $(C_{t})$ corresponds to a single curve in $(K^{*})^{2}$ .

Under this identification, the limit of the functions $\log _{t}$ as $t\rightarrow \infty$ is given by simply picking off the highest nonzero power of $t$ . We call this ${\text{val}}$ , so that:

{\begin{aligned}{\text{val}}:\ &K^{*}\rightarrow \mathbb {R} \\&\sum _{q\in \mathbb {Q} }a_{q}t^{q}\mapsto \max\{q:a_{q}\neq 0\}\end{aligned}} In analogy with our earlier constructions, we define:
${\text{Val}}={\text{val}}^{2}:(K^{*})^{2}\rightarrow \mathbb {R} ^{2}$ Thinking now of $(C_{t})$ as a curve in $(K^{*})^{2}$ , we see that ${\text{Val}}(C_{t})$ coincides with the tropicalisation which we defined earlier as the limit of $\varphi _{t}(C_{t})$ .

Thus we can redefine a tropical curve as being the image under ${\text{Val}}$ of an algebraic curve in $(K^{*})^{2}$ .