The Stacks project

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?

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 ...

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