Indeed, there is reason to suspect that difficulties could increase with increasing $n$.
We suspect that our methods should extend to offer at least some description of characteristic factor-tuples for larger numbers of commuting transformations.
While we suspect that the methods of the present section can be brought to bear on this question also, we will not pursue this analysis in detail here.
Unfortunately, the details of the calculations were omitted, and there is some doubt on whether the result is correct since our analysis suggests that $P_2$ must vanish to third order; the presence of $L^{-2}$ is also suspect.
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