Computing the degree of a finite morphism $\mathbb{P}^n\to \mathbb{P}^n$

Solution 1:

Sketch of alternative: Consider the ring map $k[Y_0, \ldots, Y_n] \to k[X_0, \ldots, X_n]$, $Y_i \mapsto f_i(X_0, \ldots, X_n)$. Looking at Hilbert functions (thinking of these as graded rings) we see that $k[\underline{X}]$ has rank $d^{n + 1}$ as a $k[\underline{Y}]$-module (hint: compute leading coefficients of Hilbert pols and use that a graded module supported in a proper closed subscheme of $\mathbf{A}^{n + 1}$ has Hilbert pol of lower degree). The degree of $f$ is the degree of the field extension $k(Y_i/Y_0) \subset k(X_i/X_0)$. The reason this is $d^n$ and not $d^{n + 1}$ is because we are taking degree $0$ parts in the extension $k(Y_i/Y_0)[Y_0^{\pm 1}] \subset k(X_i/X_0)[X_0^{\pm 1}]$ which has degree $d^{n + 1}$ by the above.