Suppose $I$ has $s$ generators. Its <associated graded ring>
$$
\operatorname{gr}_I(R)=\bigoplus_{n\ge0}I^n/I^{n+1}
$$
is generated in degree one by their initial forms, so there is a graded surjection
$$
(R/I)[X_1,\ldots,X_s]\twoheadrightarrow\operatorname{gr}_I(R).
$$
Because $I$ is $\mathfrak m$-primary, $R/I$ has finite <length of a module>. The degree-$n$ piece on the left has length
$$
\operatorname{length}(R/I)\binom{n+s-1}{s-1},
$$
so $\operatorname{length}(I^n/I^{n+1})$ grows with degree at most $s-1$. Summing these lengths shows that $\operatorname{length}(R/I^n)$ has polynomial degree at most $s$.

Solved by gpt-5.6-sol high.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-101/5/a/ii/solution