if they belong to the same conjugacy class (see [@Duval89] and [@verger]).

Infinity Branches {#sec-Infinity branches}
=================

In this section, we introduce the notion of [*infinity branch*]{} (see Definition \[D-infinitybranch\]), and we obtain some properties concerning to these algebraic entities.

For this purpose, we consider an algebraic affine plane curve $\cal C$ over $\mathbb{C}$, defined implicitly by the irreducible polynomial $f(x,y) \in {\Bbb R}[x,y]$. Let ${\cal C}^*$ be its corresponding projective curve defined by the homogeneous polynomial $F(x,y,z) \in {\Bbb R}[x,y,z]$. Furthermore, let $P=(1:m:0),\,m\in {\Bbb C},$ be an infinity point of ${\cal C}^*$, and we consider the curve defined implicitly by the polynomial $g(y,z)=F(1:y:z)$. Observe that $g(p)=0,$ where $p=(m,0)$.

By applying Theorem \[T-Puiseux\], we compute the series expansion for the solutions of $g(y,z)=0$. There exist exactly $\deg_{Y}(g)$ solutions given by different Puiseux series that can be grouped into conjugacy classes. Let one of these solutions be given by the following Puiseux series: $$\varphi(z)=m+a_1z^{N_1/N}+a_2z^{N_2/N}+a_3z^{N_3/N} +
\cdots\in{\Bbb C}\ll z\gg,\quad a_i\not=0,\, \forall i\in {\Bbb
N},$$ where $\nu(\varphi)=N\in {\Bbb
N}$, $N_i\in {\Bbb N},\,\,i=1,\ldots$, and $0<N_1<N_2<\cdots$. We have that $g(\varphi(z), z)=0$ in some neighborhood of $z=0$ where $\varphi(z)$ converges. Then, there exists some $M \in {\Bbb R}^+$ such that $$F(1:\varphi(t):t)=g(\varphi(t), t)=0,\quad \mbox{for\, $t\in {\Bbb C}$\, 