Suppose $f \in \mathbb{C}[x_1,..., x_n]$ is such that $f(\lambda^{q_1}x_1,..., \lambda^{q_n}x_n) = \lambda^df(x_1,..., x_n)$. Is there a natural grading on $\mathbb{C}[x_1,..., x_n]/(f)$?
If $q_i = 1$ for all $i$ and $d = n$, then the answer is clearly yes, but is this true in general?