# 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

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:

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