Reverse engineered Diophantine equations over $\mathbb{Q}$

Santicola, Katerina

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 897-904

Let ${𝒫}_{\mathbb{Q}}=\{{\alpha }^n:\alpha \in \mathbb{Q},n\ge 2\}$ be the set of rational perfect powers, and let $S$ be a finite subset of ${𝒫}_{\mathbb{Q}}$. We prove the existence of a polynomial $f_S\in \mathbb{Z}\left[X\right]$ such that $f\left(\mathbb{Q}\right)\cap {𝒫}_{\mathbb{Q}}=S$. 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.

Soit ${𝒫}_{\mathbb{Q}}=\{{\alpha }^n:\alpha \in \mathbb{Q},n\ge 2\}$ l’ensemble des puissances parfaites rationnelles, et soit $S$ un sous-ensemble fini de ${𝒫}_{\mathbb{Q}}$. Nous prouvons l’existence d’un polynôme $f_S\in \mathbb{Z}\left[X\right]$ tel que $f\left(\mathbb{Q}\right)\cap {𝒫}_{\mathbb{Q}}=S$. Ceci généralise un théorème récent de Gajović qui a démontré un résultat similaire pour les sous-ensembles finis de puissances parfaites entières. Notre approche fait appel à la résolution de l’équation de Fermat généralisée de signature $(2,4,n)$ dans [2, 4, 7], ainsi qu’à la finitude des puissances parfaites dans les suites récurrentes binaires non dégénérées, prouvée par Pethő et par Shorey et Stewart.

Classification : 11D41, 11D61
Keywords: Diophantine equations, rational points

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