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Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by:

$\sigma_k: X \mapsto X + S + X^k$

$\sigma_k: S \mapsto S + S^3$.

Conjecture: There exists a principal ideal $\langle a\rangle$ other than $\langle S\rangle$ such that $\langle a\rangle$ is stable by $\sigma$. Namely $\sigma(a) = ab$ with some element $b$ of $\mathbb{F}_3[[X,S]]$.

$\mathbb{F}_3[[X,S]]$ is a $2$-variable power series ring over the finite field $\mathbb{F}_3$, but I guess $\langle a\rangle$ could be found somehow.

I tried to apply Brouwer's fixed point formula somehow, but already $\langle S\rangle$ occupies one fixed point in Brouwer's theorem.

Please just help me with any of your ideas.

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  • $\begingroup$ S^2? [filler filler] $\endgroup$
    – eric
    Nov 28, 2013 at 7:26
  • $\begingroup$ Yes, but we should omit (S^k), which should be obvious. $\endgroup$
    – Pierre
    Nov 28, 2013 at 11:06

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