I have given a finite set $S$ of polynomes in the ring $R = C[x_1,\dots,x_n]$. I need to know the minimal group $G$ wich acts on $R$ such that $C[S]$ is the ring of invariants of $R$ under the action of $G$. In other words: $C[S] = R^G$.

for example: let $R = C[x_1,x_2,x_3]$ and $R^G = C[ x_1 x_2 x_3, x_{1}^3,x_{2}^3,x_{3}^3 ]$

let $G \subset SL_3(C)$ act via $a e_i \mapsto a x_i$. If $\xi$ is a third primitive root of unity, then $G$ must be generated by the diagonal matrix($\xi,\xi,\xi$).

Does there exist an algorithem for this kind of problem? I just know some the other way round - given the group, looking for the invariants.

greatz Johannes

edit: rewritten the problem to remove ambiguities because i mixed up ideals and rings in a very carelessly way. edit2: fixed the example.