Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x\_1,...,x\_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$).
Now the symmetric group $\mathfrak{S}\_n$ acts by permutation on the indeterminates.
The subring of invariant polynomials $R^{\mathfrak{S}\_n}$ has a nice description (by generators and relations) in terms of symmetric functions.

What if I only consider the action of the cyclic group $Z\_n$? 
Does anyone know if the ring $R^{Z\_n}$ admits a nice presentation?
(in the case at hand I had $\mathbb{C}[x\_1,x\_2,x\_3]$ and the action of the cyclic group $Z\_3$. Maybe in this case we can use some formula (assuming there is one) for groups splitting as semidirect products?)