Definition 10.58.3. Let $A$ be an abelian group. We say that a function $f : n \mapsto f(n) \in A$ defined for all sufficient large integers $n$ is a numerical polynomial if there exists $r \geq 0$, elements $a_0, \ldots , a_ r\in A$ such that

$f(n) = \sum \nolimits _{i = 0}^ r \binom {n}{i} a_ i$

for all $n \gg 0$.

Comment #3404 by Jonas Ehrhard on

By definition we have $f(n) = \sum_{i = 0}^ r \binom {n}{i} a_ i$ $\forall n\gg 0$. And $f(n)$ is also only defined for all $n\gg 0$. Does this mean that from the point where $f$ is defined, it equals the polynomial or might there be a second bound?

Comment #3407 by Fan on

No. Otherwise Proposition 10.58.7 won't be correct as stated.

Perhaps a better wording is: "a partial function $f:n\mapsto f(n)\in A$ is a numeric polynomial if for all sufficiently large integers $n$, $f(n)$ is defined, and there exists $r\ge0$, elements $a_0,\dots,a_r\in A$ such that ...

Comment #3466 by on

OK, I think I am going to leave it as is until there are more complaints about this.

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• 5 comment(s) on Section 10.58: Noetherian graded rings

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