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For an integer $m > 1$, let us define the action
$$ f: X_i \to (1+X_i)^{m} - 1 $$ on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ideal $I$ in $C[[X_1,...,X_N]]$.

Question: For which $V(I)$ is it possible that $V(I)$ is stable under the action $f$? Equivalently for which $I$ do we have that $f(I) = I$?

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  • $\begingroup$ I made some syntactical changes to your question and formatted formulas to make it clearer. If I misunderstood something, feel free to re-edit. $\endgroup$
    – Joonas Ilmavirta
    Commented Aug 28, 2014 at 11:48
  • $\begingroup$ Why is $f$ an action? $f$ takes a non-unit to a unit! $\endgroup$
    – Mohan
    Commented Sep 7, 2014 at 15:45

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