# Invariant Polynomes under group action - given the invariants looking for the group. algorithmic solution?

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.

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Ideals rarely contains $1$ but $1$ always lie in the invariant ring so I don't understand the question. –  Torsten Ekedahl Aug 15 '11 at 10:12
I rewrote the problem, because i mixed up ideals and rings in a very carelessly way. I did not think about constant polynomes being invariant under the action when I wrote down the problem the first time. Thnx Torsten for remind me about this basics ;) greatz Johannes. –  Johannes Aug 15 '11 at 22:08
Sorry but now your example is still wrong, we have that $x_1x_2x_3$ is also invariant under your scalar matrices (you have to replace the scalar matrices with all diagonal matrices whose non-zero entries are third roots of unity). –  Torsten Ekedahl Aug 16 '11 at 4:26
That is a case wich could appear. I fixed the example, so that the Group and the subring fit together. –  Johannes Aug 16 '11 at 8:54