# Puiseux Series

In Section The Graph of an Amoeba we introduced in a rather ad hoc way families of curves ${\displaystyle (C_{t})}$, 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 ${\displaystyle (C_{t})}$ is far more natural: namely, we think of this family of curves in ${\displaystyle (\mathbb {C} ^{*})^{2}}$ as being a single curve in ${\displaystyle K^{2}}$, where ${\displaystyle K}$ is the field of Puiseux series.

The Puiseux series are defined to be formal power series (with coefficients in ${\displaystyle \mathbb {C} }$) indexed by ${\displaystyle \mathbb {Q} }$

${\displaystyle \sum _{q\in \mathbb {Q} }a_{q}t^{q}}$
such that the set of ${\displaystyle q\in \mathbb {Q} }$ with ${\displaystyle a_{q}\neq 0}$ 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 ${\displaystyle K}$ the resulting theory will be similar to that of the complex numbers.

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

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

{\displaystyle {\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:
${\displaystyle {\text{Val}}={\text{val}}^{2}:(K^{*})^{2}\rightarrow \mathbb {R} ^{2}}$
Thinking now of ${\displaystyle (C_{t})}$ as a curve in ${\displaystyle (K^{*})^{2}}$, we see that ${\displaystyle {\text{Val}}(C_{t})}$ coincides with the tropicalisation which we defined earlier as the limit of ${\displaystyle \varphi _{t}(C_{t})}$.

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