I am interested in finding a generating set of the radical of an ideal given a set of generators for the ideal itself, but after a lot of thought I cannot figure out a good way to do it. Specifically:
Let $k$ be an algebraically closed field, and $I \subset k[x_1, ..., x_n]$ an ideal. If $I = (f_1, ..., f_m)$, is there any good way to find a set of generators for $\text{rad}(I)$?
Edit: One may assume that $f_1, ..., f_m$ form a Gröbner basis for $I$. (Given any set of generators for an ideal in $k[x_1,..., x_n]$ it is always possible to find a Gröbner basis, so this assumption is without loss of generality.)