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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 $n \gg 0$.

Comments (3)

Comment #3404 by Jonas Ehrhard on

By definition we have . And is also only defined for all . Does this mean that from the point where 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 is a numeric polynomial if for all sufficiently large integers , is defined, and there exists , elements such that ...

Comment #3466 by on

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

There are also:

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

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