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

Problem. Given f∈Z[x,y],a,b∈Q,r∈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!

thebogatron
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  • 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|. – Max Alekseyev May 04 '22 at 18:53

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