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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.

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    ??? two algebras may have the same Hilbert series and *not* be isomorphic as algebras. Example: $A=k[x]/x^3$ with $x$ of degree one, $B=k[x,y]/(x^2,y^2,xy)$ with $x$ in degree one, $y$ in degree two.2011-08-23
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    Dear mt, I first thought you had made a mistake and that's why I decided to write an answer . But I hadn't seen that you slily took $y$ to have degree$2$ and so your remark is perfectly correct. Nice trick that !2011-08-23
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    This is the kind of questions that answers itself upon the consideration of a couple of examples :)2011-12-28
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    Of course, it is still very useful to know that two Poincare series are equal! Because if you have an actual (graded) map (of algebras if you want) $A \rightarrow B$ and you can show it's either injective or surjective, then it's an isomorphism by the equality of the Poincare polynomials. Very useful.2011-12-28

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