# Decidability of a polynomial-exponential equation in two variables

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!

• 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 $|a|< 1 < |b|$. May 4 at 18:53