Let $R=\Bbb{C}[[X,Y,Z]]$ then I want to compute the Krull dimension of $R$.
My idea was to compute the Samuel function and bring it into a polynomial "form" then we immediately know that the Krull dimension is the degree of the polynomial.
For the Samuel function I need to compute $m^k/m^{k+1}$ for the maximal ideal $m\in R$. I know that the maximal ideal of $R$ is $m=(X,Y,Z)$.
After computing some powers I found out that $$m^j=(X^kY^lZ^{j-k-l})_{0\leq k\leq i,~~0\leq l\leq i-k}$$But now I struggle in finding out what $m^j/m^{j+1}$ is. Can someone help me how do proceed here?