Cyclically symmetric functions Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)?
E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ring of all cyclically-symmetric polynomials in $x$, $y$, and $z$?
Web-accessible and free references would be preferred.
 A: A good reference is Richard Stanley's "Invariants of finite groups and their applications to combinatorics,"  Bull. Amer. Math. Soc. 1 (1979), 475–511. His example 3.6 is a cyclic group of order 4.
A: Consider the action of the cyclic group $G$ of order $n$ acting on an $m$-dimensional vector space $V$.  I'll assume you are working over an algebraically closed field $k$ (or at least a field containing the nth roots of unity).  Over such a field the elements of the cyclic group are diagonalizable.  Thus the group is generated by $g=\diag(\omega^{-a_1},\dots,\omega^{-a_m})$ where $\omega$ is a primitive nth root of unity and the $a_i$ are integers (which may be assumed to lie between 0 and $n-1$.   
If $\{x_1,\dots,x_m\}$ is a basis for $V^*$ dual to the diagonal basis for $V$, then $g$ acts via $g\cdot x_i = \omega^{a_i} x_i$.  Hence if $t = x_1^{e_1}\cdots x_m^{e_m}$ is a monomial then
$g\cdot t = \omega^{a_1e_1 + \cdots a_me_m} t$.  Since a function is invariant iff it is a linear combination of invariant monomials, the invariant theory reduces to solving the linear congruence
$a_1e_1 + \cdots a_me_m \equiv 0 \pmod{n}$ for $(e_1,\dots,e_m)$.  These solutions form a monoid in $N^m$.  Finding a set of minimal generators for $k[V]^G$ amounts to finding minimal generators for this monoid.  This is not really solved in general but there has been a lot of study, as you might imagine.  
I suggest the following references.
Weidong Gao, Alfred Geroldinger,
Zero-sum problems in finite abelian groups: A survey,
Expo. Math. 24 (2006) 337 – 369
John C. Harris and David L. Wehlau,
Non-Negative Integer Linear Congruences,
Indagationes Mathematicae {\bf 17} No. 1 (2006) 37-44.  arXiv:math/0409489v1
Pingzhi Yuan,
On the index of minimal zero-sum sequences over finite cyclic groups,
Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551
Finally, I'll mention that things are much much more complicated (equivalent to the invariant theory of SL(2,C)) when the characteristic of $k$ divides $n$.
