Suppose $J \subseteq \mathbb{C}[x_1,\ldots,x_n]$ is an ideal, and $I = \sqrt{J}$ is its radical. How large can the smallest integer $e$ for which $I^e \subseteq J$ be, specifically in terms of the maximal degree(s) of the Groebner basis of $J$ and $I$, and the number of variables $n$?
Is anything known in this regard?
I believe we can assume that $I$ is prime, and thus $J$ is primary. That is, knowing some upper bound that holds for all primary ideals $J$ would also give a bound for the general case.
So, maybe this could be seen as an analogue of "multiplicity of a root" for multivariates.
EDIT on 21-03-2024: Clarified that I want an upper bound in terms of degrees of Groebner bases and the number of variables. Also added why the primary ideal case could be sufficient.