Let $A$ be a Noetherian local ring of dimension $n$, $I$ be an ideal such that $\dim A/I=m$. By Krull’s principal ideal theorem $I$ cannot be generated by less than $n-m$ elements.
Is there a criterion when can $I$ be generated by exactly $n-m$ elements? When it is not possible, can the discrepancy be bounded?
Think I figured out. Let $\mathfrak{m}$ be the maximal ideal of $A$ and $k=A/\mathfrak{m}$ be the residual field. Then the minimal number of generators for $I$ is, by Nakayama, $\dim_k I/I\mathfrak{m}$. So the “criterion” is simply: $$ \dim_k I/I\mathfrak{m} = \dim A - \dim A/I. $$
There is probably no general further refinement(s). For example when $I=\mathfrak{m}$ then the above becomes $\dim A=\dim_{A/\mathfrak{m}} \mathfrak{m}/\mathfrak{m}^2$. These are regular local rings which are themselves subject of intensive study.