The Catalan Equation in Finitely Generated Domains

We consider the Catalan equation $x^p - y^q = 1$ in unknowns $x, y, p, q$, where $x, y$ are taken from an integral domain $A$ of characteristic $0$ that is finitely generated as a $\mathbb{Z}$-algebra and $p, q > 1$ are integers. We give explicit upper bounds for $p$ and $q$ in terms of the defining parameters of $A$. Our main theorem is a more precise version of a result of Brindza. Brindza also gave inexplicit bounds for $p$ and $q$ in the special case that $A$ is the ring of $S$-integers for some number field $K$. As part of the proof of our main theorem, we will give a less technical proof for this special case with explicit upper bounds for $p$ and $q$.