Suppose I have two finite-dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbb{C}$-algebras $A = \bigoplus_{k \geq 0} A_{k}$ and $B = \bigoplus_{k \geq 0} B_{k}$ with Hilbert-Poincaré series, $P_{A}(t) = \sum_{k \geq 0} \dim A_{k} \ t^{k}$ and $P_{B}(t) = \sum_{k \geq 0} \dim B_{k} \ t^{k}$, respectively.
When is it true that $A$ and $B$ are isomorphic as graded $\mathbb{C}$-algebras if $P_{A} = P_{B}$? Suppose that $P_{A} \neq P_{B}$ but $P_{A}(1) = P_{B}(1)$, what can be said about $A$ and $B$ in this case? Are they isomorphic as $\mathbb{C}$-algebras but not as graded $\mathbb{C}$-algebras?
The algebras that brought me to ask these questions are all of the form $\mathbb{C} \{ z_1, \dots, z_n \} / J$, where $J$ is a finitely generated ideal of partial derivatives of a complex analytic function $f$ with an isolated critical point at the origin.