Suppose I have a polynoial $p:\mathbb{C} \to \mathbb{C}$. There's a natural way to extend this polynomial to a polynomial $p:\mathbb{CP}^1 \to \mathbb{CP}^1$. If you have a polynomial $p(z) = a_nz^n + a_{n-1}z^{n-1} + \dotsb + a_0$, with $a_n \neq 0$, then let $$ \hat{p}([z : w]) = \left[ \sum_{k = 0}^n a_k z^k w^{n-k} : w^n \right]. $$
Okay, after that I proved that a polynomial $p(z) = a_nz^n + a_{n-1}z^{n-1} + \dotsb + a_0$ is fixed by a finite amount of values (using a vandermonde matrix). I'm trying to prove the fundamental theorem of algebra. But for this I first need to prove that $\hat{p}$ has finite singular values (using the fact that $p$ is fixed by a finite amount of values). How is this done?