# Compute the Hilbert-Samuel function

Let $$f\in R=k[x,y,z]_{(x,y,z)}$$ be a homogeneous polynomial of degree $$d$$, monic in $$x$$. Show that $$(y,z)$$ is an ideal of finite colength on $$M=R/(f)$$. Compute the corresponding Hilbert-Samuel function.

Maybe someone can do it as an example.