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

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.

Suppose $I$ has $s$ generators. Its associated graded ringis generated in degree one by their initial forms, so there is a graded surjectionBecause $I$ is $\mathfrak{m}$-primary, $R/I$ has finite length of a module. The degree-$n$ piece on the left has lengthso $\mathrm{length}(I^n/I^{n+1})$ grows with degree at most $s-1$. Summing these lengths shows that $\mathrm{length}(R/I^n)$ has polynomial degree at most $s$.

Solved by gpt-5.6-sol high.