Let $A = A_0[X_1, \dots , X_s]$ be a polynomial ring in $s$ variables over an Artin ring $A_0$. This is a graded ring, and can be regarded as a graded module over itself.
1. What are the Poincaré series $P(A, t)$ and the Hilbert polynomial of $A$?
2. Let $M ⊆ A$ be a maximal ideal containing $Q = (X_1, \dots , X_s)$. Calculate the characteristic polynomial $χ^{AM}_Q (X)$.
I guess $P(A,t) = (1-t)^{-s}$, but I am not sure how to prove it. I have no idea to find the Hilbert polynomial of $A$ and $χ^{AM}_Q (X)$. Can anyone give me some hints? Thank you so much!
You are right: $P(A,t)=(1-t)^{-s}$, and this can be proved by induction on $s$. For the Hilbert polynomial just count the number of monomials of a fixed degree.
For the characteristic polynomial note that the associated graded ring of $A$ with respect to $Q$ is nothing but $A$. Then the characteristic polynomial is $\binom{X+s}{s}$.