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Say I have a cluster algebra with principal coefficients and initial cluster $x_1,\ldots,x_n$. I don't want to invert the coefficient variables $y_1,\ldots,y_n$. The Laurent Phenomenon says that every cluster variable (and thus every element of the cluster algebra) is a Laurent polynomial in the $x_i$ with coefficients integer polynomials in the $y_i$.

The question is simple: If I hand you an expression for a Laurent polynomial in the $x_i$ with coefficients integer polynomials in the $y_i$, can you tell me if it is an element of the cluster algebra?

Answers might take the form of a criterion that could be checked by hand or an algorithm. I would also be interested in algorithms or partial criteria that can provide answers in some cases but not all cases.

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    $\begingroup$ Thinking a bit more about this: It occurs to me that it's easy to test membership in the upper cluster algebra, so in the case where the cluster algebra equals the upper cluster algebra, this question should be easy. I'll try to think more about this and make sure I'm not missing something and then post an answer. (I may soon be wondering what the state of the art is on when the cluster algebra equals the upper cluster algebra!) In any case, I am still curious about testing membership in the general case. $\endgroup$
    – Nathan Reading
    Commented Sep 19, 2014 at 18:56

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