# Poincaré series and the Hilbert polynomial of $A = A_0[X_1,\dots , X_s]$ [closed]

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!

## closed as off-topic by user26857, user91500, Watson, choco_addicted, Daniel W. FarlowJun 21 '16 at 10:58

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1. 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.
2. 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 $\binom{X+s}{s}$.