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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 101 / 5 / a / i / Solution

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{wiki=Hilbert–Samuel_function}

For an $\mathfrak m$-primary ideal $I$ in a Noetherian local ring, $\operatorname{length}(R/I^n)$ agrees for all sufficiently large $n$ with the Hilbert-Samuel polynomial of $I$.


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 Incoming links: Hilbert-Samuel polynomial