Puiseux Series

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

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

$$\sum _{q\in \mathbb {Q} }a_{q}t^{q}$$

$$\sum _{q\in \mathbb {Q} }a_{q}t^{q}$$

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

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

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

$${\begin{aligned}{\text{val}}:\ &K^{*}\rightarrow \mathbb {R} \\&\sum _{q\in \mathbb {Q} }a_{q}t^{q}\mapsto \max\{q:a_{q}\neq 0\}\end{aligned}}$$

$${\begin{aligned}{\text{val}}:\ &K^{*}\rightarrow \mathbb {R} \\&\sum _{q\in \mathbb {Q} }a_{q}t^{q}\mapsto \max\{q:a_{q}\neq 0\}\end{aligned}}$$

In analogy with our earlier constructions, we define:

$${\text{Val}}={\text{val}}^{2}:(K^{*})^{2}\rightarrow \mathbb {R} ^{2}$$

$${\text{Val}}={\text{val}}^{2}:(K^{*})^{2}\rightarrow \mathbb {R} ^{2}$$

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

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