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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.

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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.

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    $\begingroup$ 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.) $\endgroup$ Jun 4 at 16:30
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    $\begingroup$ Yes, the equation is true if and only if the invariant ring is Cohen-Macaulay, which was implicitly assumed in Schemer's question. If it's not Cohen-Macaulay, I'm not aware of a realistic upper bound for s. A rediculously large bound can be derived from Symond's degree bound. $\endgroup$ Jun 4 at 16:55

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