Let $\mathbb{P}^1$ be the complex projective line and $\mathbb{P}^2$ be the complex projective plane. Let $\mathcal{C} \subset \mathbb{P}^2$ be a compact projective plane curve, i.e. the zero locus of a homogeneous polynomial $F \in \mathbb{C}[X,Y,Z]$ of degree $d$. We consider the holomorphic map between Riemann surfaces $\pi : \mathcal{C} \to \mathbb{P}^1, \, [x:y:z] \mapsto [x:z]$. I want to show that the degree of this map is equal to $d$, as it is stated in "Algebraic Curves and Riemann Surfaces" by R. Miranda (Plücker's formula, page 144) and in this post (without further details) : Genus of a smooth projective curve.
My attempt is the following. We take $p = [x_0,z_0] \in \mathbb{P}^1$ such that it is not the image by $\pi$ of a ramification point (it is possible as ramifications points are a finite subset of $\mathcal{C}$, the latter being compact). Then the degree of $\pi$ is just the number of preimages of $p$. Without loss of generalities we suppose $x_0 \neq 0$ so we can rewrite $p = [1:w_0]$. Now a preimage of $p$ is of the form $[1:y:w_0]$ and such that $F(1,y,w_0) = 0$. If $f(Y) = F(1,Y,w_0) \in \mathbb{C}[Y]$ we see that preimages of $p$ are in bijection with roots of $f$.
So I'd like to show that $f$ has $d$ distinct roots, but I don't see how to prove this (and I'm not 100% sure it is the right statement). Any help is welcome !
