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My question is with regards to the following (algorithmic) problem:

Problem. Given $f\in \mathbb{Z}[x,y], a,b\in \mathbb{Q}, r\in \mathbb{Z}$, do there exist positive integers $m,n$ such that $f(m,n) = r a^m b^n$?

Is this problem decidable? Is it decidable in any special case (e.g. taking a specific (non-trivial) $f$, taking $a,b\in \mathbb{Z}$ or choosing specific $a,b,r$)?

Thank you!

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  • $\begingroup$ Apparently, if both $|a|,|b|>1$ or both $|a|,|b|<1$, then there is only a finite number of candidate $m,n$. So, interesting case is when say $|a|< 1 < |b|$. $\endgroup$ Commented May 4, 2022 at 18:53

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