# Are there any results on an upper bound for the number of secondary invariants needed to generate the invariant ring of a finite group?

If $$G$$ is a finite cyclic group, $$\beta: G \to \operatorname{GL}(\mathbf{V})$$ is a linear $$n$$ dimensional representation of $$G$$, and $$\{x_{1},\dots,x_{n}\}$$ is a basis of $$\mathbf{V}^{\ast}$$. If $$\{f_{1}(X),\dots,f_{n}(X)\}$$ are a homogeneous system of parameters for the ring $$k[X]^{G}$$, then the $$f_{i}(X)$$ are called primary invariants. If $$\{f_{1}(X),\dots,f_{n}(X)\}$$ are a collection of $$n$$-primary invariants, $$A$$ is the ring $$k[f_{1}(X),\dots,f_{n}(X)]$$ and $$g_{1}(X),\dots,g_{s}(X)$$ are homogeneous invariant polynomials such that $$k[X]^{G} = \oplus_{i=1}^{s} A g_{i}(X)$$, then the invariants $$g_{i}(X)$$ are called secondary invariants. Are there any results for upper bounds on the number of secondary invariants in a minimal set of secondary invariants? A satisfactory result would involve a set of conditions on the characters of this representation.

Update: I have to make a clarification about the application which will describe what a satisfactory result will look like. Suppose that we wanted to describe congruence properties of a sequence of natural numbers $$\{a_{i}\}_{i=1}^{\infty}$$ modulo $$M$$. Let $$k[t]_{t}/\langle t^{M}-1 \rangle$$ be the affine coordinate ring of $$\mu_{M}$$ and $$\{x_{1},\dots,x_{n}\}$$ be a basis of a vector space $$\mathbf{V}^{\ast}$$. Suppose that a number in this sequence has the desired property if and only if $$\overline{f_{i}(a_{i})} \equiv \overline{b_{i}} \pmod{M}$$, for some sequence $$\{b_{i}\}_{i=1}^{\infty}$$, then one can define an $$n$$-dimensional representation $$\beta: \mu_{M} \to \operatorname{GL}(\mathbf{V})$$, via the co-action which sends $$x_{i}$$ to $$\overline{t}^{f_{i}(a_{i})-b_{i}} x_{i}$$ for $$1 \le i \le n$$. Suppose in addition that we want to prove that an infinite sub-sequence of such numbers exists. If this is our goal, then it is necessary and sufficient to show that for every $$N \in \mathbb{N}$$ there is an $$n \in \mathbb{N}$$ such that $$\dim\left(\left(\mathbf{V}^{\ast}\right)^{\mu_{M}}\right)>N$$ (here this refers to the linear $$\mu_{M}$$-invariant polynomials) for a vector space of dimension $$n$$. If this is not true, then there is some $$L \in \mathbb{N}$$ such that only $$L$$ polynomials of the primary invariants may be linear. By Theorem 3.7.1 in Computational Invariant Theory by Derksen and Kemper, \begin{align*} s&= \left(\prod_{i=1}^{n} \deg(f_{i}(X))\right)/M \\ &\ge 2^{n-L}. \end{align*} If there was an upper bound related to the weights of the characters, then a proof would be possible in some cases. One could try to use asymptotic estimates of the upper bound and lower bound to prove that as $$n \to \infty$$ the upper and lower bound conflict. Such a contradiction would mean that an infinite sequence of linear invariant polynomials exists, which in turn would mean that an infinite sub-sequence of $$\{a_{i}\}_{i=1}^{\infty}$$ with the desired property would exist. The characters would be related to the sequence of numbers. Thank you @Gregor Kemper for mentioning that I did not mention that I needed a bound not related to Theorem 3.7.1 in your book.

By substituting $$G$$ with $$G/$$ker$$(\beta)$$, you may assume that $$\beta$$ is injective. Then the number of secondary invariants is given by the formula $$s = |G|^{-1} \prod_{i=1}^n \deg(f_i)$$. This can be found in Theorem 3.9.1. from "Computational Invariant Theory" by H. Derksen and G. Kemper.
• Is this always true if the characteristic of $k$ divides the order of $G$? (Your book isn't immediately available, but I found a survey paper of yours that indicates one needs the invariant ring to be Cohen-MacCaulay.) Jun 4 at 16:30