I found a theorem about multigraded Hilbert series stated as follows:


Let $R$ be a Noetherian multigraded algebra $R:=\bigoplus\_{j\in\mathbb{N}^m}{R\_j}$ over $R\_0=\mathbb{C}$. If $R$ is generated by $h$ homogeneous polynomials of multidegrees $\alpha\_1,\dots,\alpha\_h$, then its Hilbert series $HS\_R(\mathbf{t})$ can be expressed as $$HS\_R(\mathbf{t})=\frac{P(\mathbf{t})}{\prod\_{\ell=1}^h{(1-\mathbf{t}^{\alpha\_\ell})}},$$ where $P(\mathbf{t})$ is a polynomial with integer coefficients. Furthermore, if $R$ is freely generated, then $P(\mathbf{t})=1$. (here $\mathbb{t}^{\alpha}=t\_1^{\alpha\_1}\cdots t\_n^{\alpha\_n}$).


I can no longer find the place where I read this. If anyone could provide me with a reference, that would be greatly appreciated.



I need to refind a reference on multigraded Hilbert series
As @Richard points out, this follows easily from proof with m = 1. In particular, you can do an induction on h and use the fact that length is additive on short exact sequences. However if you are looking for a specific reference, [this](http://www.math.lsa.umich.edu/~hderksen/Publications/denom2005.pdf) paper might be of interest, but note again as @Richard said, the proof is to just write down Hilbert's original argument and see that it carries over. I hope this helps. 


