Let $A$ be a $\mathbb{C}$-algebra, generated by $n$ elements ($n$ finite). Assume that $A$ has a unique minimal prime ideal $\mathfrak{p}$. Write $t$ for the minimal number of generators for the ideal $\mathfrak{p}$.
Can we bound $t$ in terms of $n$?
I assume this is either false, or a special case of something much more general. In the latter case, maybe there's a simple proof for this special case? I'm not asking for the best possible bound (but it would be nice if the bound is pretty good).
There are examples by Macaulay of prime ideals of height two in polynomial ring $R$ in three variables over a field which require large number of generators. If $P$ is one such, taking the ring $R/P^{(n)}$ for large enough $n$, you will have such examples showing no such bound can exist.