is a set $\displaystyle
    B=\bigcup_{i=1}^NL_i$, where $L_i=\{(z,r_i(z))\in {\Bbb C}^2: \,z\in
    {\Bbb C},\,|z|>M\}$, $i=1,\ldots,N$, $M\in {\Bbb R}^+$, and $r_1,\ldots,r_N$ are the conjugates of $$r(z)=mz+a_1z^{1-N_1/N}+a_2z^{1-N_2/N}+a_3z^{1-N_3/N} + \cdots,\quad a_i\not=0,\, \forall i\in {\Bbb N}$$ $N,N_i\in {\Bbb N},\,\,i=1,\ldots$, and $0<N_1<N_2<\cdots$.

-   An [*infinity branch of ${\cal C}$ of type 2*]{} associated to the infinity point $P=(m:1:0),\,m\in {\Bbb C}$, is a set $\displaystyle
    B=\bigcup_{i=1}^NL_i$, where $L_i=\{(r_i(z),z)\in {\Bbb C}^2: \,z\in
    {\Bbb C},\,|z|>M\}$, $i=1,\ldots,N$, $M\in {\Bbb R}^+$, and $r_1,\ldots,r_N$ are the conjugates of $$r(z)=mz+a_1z^{1-N_1/N}+a_2z^{1-N_2/N}+a_3z^{1-N_3/N} + \cdots,\quad a_i\not=0,\, \forall i\in {\Bbb N}$$ $N,N_i\in {\Bbb N},\,\,i=1,\ldots$, and $0<N_1<N_2<\cdots$.

1.  In the following, we assume w.l.o.g that the given algebraic plane curve $\cal C$ only has type 1 infinity branches; that is, all the infinity points are of the form $(1:m:0)$, $m\in {\Bbb C}$. Otherwise, we may consider a linear change of coordinates.

2.  By abuse of notation, we will say that $N$ is [*the ramification index*]{} of the branch $B$, and we will write it as $\nu(B)=N$. Note that $B$ has $\nu(B)$ leaves.

In the following example, we compute the infinity branches for a given plane curve.

\[Ej-ramas infinitas\] Let ${\cal C}$ be the plane curve defined implicitly by the irreducible polynomial $$f(x,y)=y^5-4y^4x+4y^3x^2+2y^2x-y^2x^2+2yx^2+2yx^3+x+x^2\in {\Bbb R}[x,y].$$ The corresponding projective curve ${\cal 