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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?)

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4 Answers

up vote 14 down vote accepted

The actions of $S_n$ and $\mathbb Z_n$ differ in the sense that in the first case the quotient is smooth (it is again $\mathbb C^n$) while in the second case it is singular. This is why in the fist case we have a nice presentation, but in the second not really. For example, the number of generators of the quotient can not be less than the dimension of Zariski tangent space to the singularity at zero of $\mathbb C^n/\mathbb Z_n$.

Still in principle the presentation can be provided by toric geometry (http://www.cs.amherst.edu/~dac/toric.html) because the quotient is the toric singularity. For example, in your case of $\mathbb C^3/\mathbb Z_3$ let us change the coordinates so that $\mathbb Z_3$ is acting as $w_0\to w_0$, $w_1\to \mu w_1$, $w_2\to \mu^2 w_2$ (here $\mu^3=1$). Then you can write the minimal set of four generators:

$w_0, w_1^3, w_2^3, w_1w_2$, and one obvious relation $(w_1^3w_2^3)=(w_1w_2)^3$

The case $\mathbb C^n/\mathbb Z_n$ for $n>3$ will be more involved, but the idea is the same roughly. First you chose the coordinates on $\mathbb C^n$ $w for which the action is diagonal. Then pick the minimal set of monomials (in these new coordinates) that are invariant under the action, and generate the whole set of invariant monomials (of positive degree).

Consider one more case $n=4$, and chose the coordinates $w_i$, so that $Z_4$ is acting as $w_i\to \mu^iw_i$, $\mu^4=1$. The number of generators is $7$ this time:

$w_0, w_1^4, w_3^4, w_2^2, w_1w_3, w_1^2w_2, w_3^2w_2$

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Note however that diagonalisation is not possible over $\mathbb Z$ (or even $\mathbb Q$) which make things trickier. However, my guess is that the easiest way is to first do it over a splitting field and then descend (at least to $\mathbb Q$, over $\mathbb Z$ there should be even more complications). –  Torsten Ekedahl Jul 28 '11 at 5:39
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The minimal number of invariants needed to generate ${\mathbb Z}[x_1,...,x_n]^{{\mathbb Z}_n}$ has been considered by a number of authors including Erdos, Dixmier and Kac. (see the references in [John C. Harris and David L. Wehlau Non-Negative Integer Linear Congruences, Indagationes Mathematicae 17 No. 1 (2006) 37-44]. It is easily seen to be bounded below by the number of partitions of n, ${\mathcal P}(n)$. Dixmier produced a number of papers giving the asymptotic behavior of this number as a function of $n$. The results in the above Harris-Wehlau paper are completed by using the main result in [Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551]. These two papers combine to show that the number of homogeneous generators of this ring of invariants of degree $k$ is exactly $\phi(n){\mathcal P}(n-k)$ if $k \geq \lfloor n/2\rfloor + 2$ (here $\phi$ is Euler's totient function). Surprisingly (at least to me) much less is known about the number of generators in lower degrees.

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By the general theory of invariants of finite groups, there exist $(n-1)!$ homogeneous polynomials $u_1, \dots, u_p$ ($p=(n-1)!$) such that every element $f$ of $R^{Z_n}$ can be uniquely written $f = u_1 g_1 + \cdots+ u_p g_p$, where $g_1,\dots,g_p$ are symmetric functions. I don't know whether an explicit description of $u_1,\dots,u_p$ is known for arbitrary $n$.

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I recommend that you read the chapter of this book http://www.springerlink.com/content/n7163w70l6472421/ entitled Algorithms and Invariants. In particular it shows that you you can use Groebner bases to calculate generators for the ring of invariants under the action of a finite group.

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