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Here is the question I am trying to understand it is solution:

(Poincare series of a graded algebra) Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the formal series $$P(A) = \sum_{i \geq 0} \operatorname{dim}(A_i)t^i.$$

Prove that $$P(k\{x_1, \dots , x_n \}) = \frac{1}{1 - nt} \text{ and } P(k [x_1, \dots , x_n ]) = \frac{1}{(1 - t)^n}$$

Graded algebra and Hilbert–Poincaré series, closed formula, need some more detail.

I am almost ok with the proof of the first part (the only thing that confuses me is why $|nt|<1$ ) any help will be appreciated in this case) but I do not see the dimension of The dimension of the commutative graded algebra $k[x_1, \dots, x_n]$, could someone explain this to me please?

Intuition
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1 Answers1

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"The dimension of the commutative graded algebra" $A=k[x_1, \dots, x_n]$ is $\infty.$

The dimension of its component $A_i$ of degree $i$ was not needed to compute $P_A$.

However, its computation is classical as well: since it is the number of $n$-tuples $(e_1,\dots,e_n)$ of integers $e_j\ge0$ such that $\sum e_j=i$, $$\dim A_i=\binom{n+i-1}i.$$

Anne Bauval
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