 Solution

For an $\mathfrak{m}$-primary ideal $I$ in a Noetherian local ring, the functionagrees 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.