----- > [!proposition] Proposition. ([[Hilbert Polynomial]]) > Recall the 'canonical example' in [[Poincare Series and Hilbert-Serre]]: > > ![[Poincare Series and Hilbert-Serre#^canonical-example]] > > > In that example, we can view the coefficients $\ell(A_{n})={n+s-1\choose s-1}=$ as a polynomial $p(n) \in \mathbb{Q}[n]$ in $n$ over $\mathbb{Q}$ of degree $d(M)-1$, where $d(M)=s$ is the order of the pole at $T=1$ of $R(T)$. > > This can be abstracted to any setting where we have a [[graded module]] $M$ over a [[graded ring]] $A$ generated as an $A_{0}$-algebra by degree-1 elements (i.e. $k_{1}=\dots=k_{s}$). In such a case, there is a unique polynomial $\text{HP}_{M} \in \mathbb{Q}[T]$ of degree $d(M)-1$ such that $\ell(M_{n})=\text{HP}_{M}(n)$ for all large enough $n$. > > The function $n \mapsto \ell(M_{n})$ is the **Hilbert function** of the [[graded module|graded]] $A$-[[module]] $M$. The [[polynomial 4|polynomial]] $\text{HP}_{M}$ is the **Hilbert polynomial** of $M$. > [!proof]- Proof. ([[Hilbert Polynomial]]) > ~ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```