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 ∈ℤX 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.