[*center*]{} of the place.

In the following definition, we introduce the notion of [*branch*]{} of a plane curve.

Given a local parametrization $(X,Y)$ of a plane curve $\cal C$, the set of all points $(X(t),Y(t))$ obtained by allowing $t$ to vary within some neighborhood of $0$ where $X(t)$ and $Y(t)$ converge is called a [*branch*]{} of $\cal C$.

It can be shown that two equivalent local parametrizations provide the same branch. Therefore, one obtains a branch for each place of a given algebraic plane curve.

One may prove that the center of a local parametrization of $\cal C$ is a point on $\cal C$. Conversely, from the following theorems, we also obtain that every point on $\cal C$ is the center of at least one place of $\cal C$ (see Theorems 2.77 and 2.78 in [@SWP]).

\[T-Puiseux\] [**(Puiseux’s Theorem)**]{} The field $K\ll x\gg$ is algebraically closed.

A proof of Puiseux’s Theorem can be given constructively by the Newton Polygon Method (see e.g. Section 2.5 in [@SWP]). This method solves the construction of solutions of non-constant univariate polynomial equations over $K\ll x\gg$.

\[T-puiseux-place\] Let $\cal C$ be a plane curve defined by $f(x,y)\in {\Bbb R}[x,y]$. To each root $Y(x) \in {\Bbb 