Reverse engineered Diophantine equations over ℚ

Let 𝒫 ℚ ={α n :α∈ℚ,n≥2} be the set of rational perfect powers, and let S be a finite subset of 𝒫 ℚ . We prove the existence of a polynomial f S ∈ℤ[X] such that f(ℚ)∩𝒫 ℚ =S. This generalizes a recent theorem of Gajović who proved a similar result for finite subsets of integer perfect powers. Our approach makes use of the resolution of the generalized Fermat equation of signature (2,4,n) in [2, 4, 7], as well as the finiteness of perfect powers in non-degenerate binary recurrence sequences, proved by Pethő and by Shorey and Stewart.