Hilbert Series of $k[x_1, ... x_n]$

Question

For the algebra $A = k[x_1, \dots, x_n]$ graded by degree. How does one find the Hilbert series. For a single variable, the hilbert series is simply $1+t+t^2+\dots = 1/(1-t)$.

Answer 1

What you are looking for is $(1 + t + t^2 + \dots)^n = \frac{1}{(1-t)^n}$.

Because you can easily see that adding a variable will multiply the Hilbert serie by $1 + t + t^2 + t^3 + \dots $.

Answer 2

You want to count the number of monomials of degree $n$ obtainable from $x_1,\ldots,x_k$. This is exactly a multiset on $[k]$ with $n$ elements, and there are $$\binom{n+k-1}{k}$$ of those. On the other hand, this number is equal to $$(-1)^n \binom{-n}{k}$$ which is the coefficient of $t^n$ in $(1-t)^{-n}$, whence the Hilbert series of your ring is this rational function of $t$.