Reverse Engineered Diophantine Equations over $\mathbb{Q}$

Let ${\mathcal{P}}_{\mathbb{Q}}=\{{\alpha }^n:\alpha \in \mathbb{Q},n\ge 2\}$ be the set of rational perfect powers, and let $S\subseteq {\mathcal{P}}_{\mathbb{Q}}$ be a finite subset. We prove the existence of a polynomial $f_S\in \mathbb{Z}[X]$ such that $f(\mathbb{Q})\cap {\mathcal{P}}_{\mathbb{Q}}=S$. This generalizes a recent theorem of Gajović who recently proved a similar theorem for finite subsets of integer perfect powers. Our approach makes use of the resolution of the generalized Fermat equation of signature $(2,4,n)$ due to Ellenberg and others, as well as the finiteness of perfect powers in non-degenerate binary recurrence sequences, proved by Pethő and by Shorey and Stewart.