= Initial-form lower bound for ideal generators
{title2=$\dim_k I/\mathfrak mI\text{ bounds the generator count}$}
Let $\mathfrak m$ be a <maximal ideal> of a <ring> $A$, $k=A/\mathfrak m$, and $I\subseteq\mathfrak m^d$. Since $\mathfrak mI\subseteq\mathfrak m^{d+1}$, elements of $I$ with <linearly independent> classes in $\mathfrak m^d/\mathfrak m^{d+1}$ have independent classes in $I/\mathfrak mI$. Any generating set must span this latter <vector space>. Thus $r$ independent degree-$d$ initial forms give a lower bound of $r$ generators, without assuming that height equals the generator count.
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