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

TL;DR

This paper constructs polynomials over the integers whose rational values intersect the set of rational perfect powers exactly in a given finite subset, extending previous results from integers to rationals.

Contribution

It generalizes a recent theorem by constructing such polynomials for finite subsets of rational perfect powers using advanced number theory techniques.

Findings

Existence of polynomials with prescribed intersection with rational perfect powers.

Extension of previous integer-based results to the rational setting.

Application of the resolution of generalized Fermat equations and recurrence sequence finiteness.

Abstract

Let $\mathscr{P}_\mathbb{Q}=\{ \alpha^n \; : \; \alpha \in \mathbb{Q}, \; n \ge 2\}$ be the set of rational perfect powers, and let $S \subseteq \mathscr{P}_\mathbb{Q}$ be a finite subset. We prove the existence of a polynomial $f_S \in \mathbb{Z}[X]$ such that $f(\mathbb{Q}) \cap \mathscr{P}_\mathbb{Q}=S$. This generalizes a recent theorem of Gajovi\'{c} 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 Peth\H{o} and by Shorey and Stewart.