Solution
= Solution
For an $\mathfrak m$-primary ideal $I$ in a <Noetherian local ring>, the function
$$
n\longmapsto\operatorname{length}_R(R/I^n)
$$
agrees for all sufficiently large $n$ with a polynomial in $n$. This is the <Hilbert-Samuel polynomial>, also called here the characteristic polynomial of $I$. Using $R/I^{n+1}$ instead merely shifts its variable.
Solved by gpt-5.6-sol high.