(see [@Duval89]).

The [*order*]{} of a non-zero (Puiseux or Laurent) series $A$ is the smallest exponent of a term with non-vanishing coefficient in $A$. We denote it by $\ord(A)$. We let the order of 0 be $\infty$.

In the following, we introduce the notion of projective local parametrization for a projective plane curve (see Definition 2.69, and Lemma 2.70 in [@SWP]).

Let ${\cal C}^* \subset {\Bbb P}^2({\Bbb C})$ be a projective plane curve defined by the homogeneous polynomial $F(x,y,z) \in {\Bbb R}[x,y,z]$. Let $A^*,B^*,C^*$ be series in ${\Bbb C}((t))$ such that: (i) $F(A^*(t):B^*(t):C^*(t))=0$ (where the three series converge), and (ii) there is no $D \in {\Bbb C}((t))\setminus \{0\}$ such that $D\cdot(A^*,B^*,C^*) \in {\Bbb C}^3$. Then ${\cal P^*}=(A^*:B^*:C^*) \in {\Bbb P}^2({\Bbb C}((t))\,)$ is called a [*projective local parametrization*]{} of $\cal C^*$. In addition, one can always find such a parametrization having $\min\{\ord(A^*),\ord(B^*),\ord(C^*)\}=0$, and the point ${\cal P^*}(0)\in
\cal C^*$ is called the [*center*]{} of $\cal P^*$.

For an affine plane curve, the above notion can be stated as follows:

Let ${\cal C}$ be a real plane algebraic curve over ${\Bbb C}$ defined implicitly by the irreducible polynomial $f(x,y) \in {\Bbb R}[x,y]$. Let $A,B$ be series in ${\Bbb C}((t))$ such that: (i) 