Puiseux Series

In Section The Graph of an Amoeba we introduced in a rather ad hoc way families of curves , 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 is far more natural: namely, we think of this family of curves in as being a single curve in , where is the field of Puiseux series.

The Puiseux series are defined to be formal power series (with coefficients in ) indexed by

such that the set of with 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 the resulting theory will be similar to that of the complex numbers.

Coming back to the family of curves , we now view the parameter appearing in the equation for as the formal variable in . Thus the family corresponds to a single curve in .

Under this identification, the limit of the functions as is given by simply picking off the highest nonzero power of . We call this , so that:

In analogy with our earlier constructions, we define:
Thinking now of as a curve in , we see that coincides with the tropicalisation which we defined earlier as the limit of .

Thus we can redefine a tropical curve as being the image under of an algebraic curve in .