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I'm trying to understand how to compute the Hilbert-Samuel polynomial of a specific example. Could someone help me with an elaborate computation so that I get it...

For example, what is the HS-polynomial of $\mathbb Z[x,y,z]_{(2,x,y-1,z-2)}$ (i.e. the localization of the ring $\mathbb Z[x,y,z]$ at the maximal ideal $(2,x,y-1,z-2)$)?

Thank you all in advance and happy new year to all!!!

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In this case it's easy to determine the Hilbert-Samuel polynomial since $R=\mathbb Z[x,y,z]_{(2,x,y-1,z-2)}$ is a regular local ring of dimension $4$, so its Hilbert-Samuel polynomial is ${X+4}\choose{4}$.

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See this paper for more on calculating Hilbert polynomials. There is also a Macaulay function.

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    Bayer & Stillman paper in its original place http://www.sciencedirect.com/science/article/pii/07$47$71719290024X2013-03-15