I would like to compute the Milnor number of $f(x,y,z) = x^p+y^q+z^r-xyz$ which amounts to finding the complex dimension of the underlying vector space of $\mathbb{C}[x,y,z]/(px^{p-1}-yz,qy^{q-1}-xz,rz^{r-1}-xy)$; i.e. we quotient the polynomial ring by the ideal generated by the 1st partial derivatives of $f$.
The relations are such that if we were to choose a basis for the underlying vector space, we could choose one without mixed terms. Moreover, because $px^p = qy^q=rz^r=xyz$, we can write a basis where the highest powers of $y,z$ are $q-1$ and $r-1$ respectively since if we go higher, we can convert over to $x$. However, I don't know how to argue that in this choice of basis, the highest power of $x$ is $x^p$; i.e. the Milnor number is $p+q+r-1$. I've read that this should be the correct answer.