I am currently reading a book titled "Numerical semigroups", which can be found here. I have a question regarding a definition provided by the author. To begin, let's define $F=y^n+a_1(x)y^{n-1}+\cdots +a_n(x) \in k[x][y]=k[x, y]$, where $k$ is an algebraically closed field. We also define $h_F(u, x, y) = u^nF(x/u, y/u)$.
Now, considering $\mathbb{P}^2$, the projective 2-space over $k$, with $[0:1:0]$ denoting a point in $\mathbb{P}^2$, the author defines the following:
$F$ has one place at infinity if $h_F$ is analytically irreducible at $[0:1:0]$.
Could someone please explain this definition step by step? I tried to find another source for this definition, but I was unsuccessful. Thank you in advance.
