and $B_2$ associated to $P$.

$$\begin{array}{cc}
\psfig{figure=Ejemplo1a.eps,width=4.5cm,height=4.5cm,angle=270} &
\psfig{figure=Ejemplo1b.eps,width=4.5cm,height=4.5cm,angle=270}
\end{array}$$

In the following, we prove that any point of the curve with sufficiently large coordinates belongs to some infinity branch. For this purpose, we recall the reader that if $h$ is a complex-valued function of a complex variable, $h: {\Bbb C} \rightarrow {\Bbb C}$, we say that the limit of $h(z)$ as $z$ approaches $\infty$ is $L$, written $\displaystyle\lim_{z\rightarrow\infty}h(z)=L$, if whenever $\{z_n\}_{n\in {\Bbb N}}$ is a sequence of points with $\displaystyle\lim_{n\rightarrow\infty} z_n =\infty$, it holds that $\displaystyle\lim_{n\rightarrow\infty}h(z_n)=L$ (see e.g. [@Ahlfors] or [@conway]).

\[L-BolaMaxima\] Let $\cal C$ be an algebraic plane curve. There exists $K\in\mathbb{R}^+$ such that for every $p=(a,b)\in\mathcal{C}$ with $|a|>K$, it holds that $p\in B_p$, where $B_p$ is an infinity branch of ${\cal C}$.

**Proof:** Let us assume that the lemma does not hold, and we consider a sequence $\{K_n\}_{n\in {\Bbb N}}\subset \mathbb{R}^+$ such that $\lim_{n\rightarrow\infty}K_n=\infty$. Then, for every $n\in {\Bbb N}$ there exists a point $p_n=(a_n,b_n)\in\mathcal{C}$ such that $|a_n|>K_n$, and $p_n$ does not belong to any infinity branch of $\cal C$.

Let $P_n=(a_n:b_n:1)$. Since $F(P_n)=f(p_n)=0$, then $\lim_{n\rightarrow\infty}F(P_n)=0$. Thus, we distinguish two different cases:

1.  If there exists a monotone subsequence $\{b_{n_l}/a_{n_l}\}_{l\in {\Bbb N}}$ that is not bounded, we 