论文标题
$ \ mathbb {q} $上的反向工程二芬太丁方程
Reverse Engineered Diophantine Equations over $\mathbb{Q}$
论文作者
论文摘要
令$ \ mathscr {p} _ \ mathbb {q} = \ {α^n \; ：\; α\ in \ mathbb {q}，\; n \ ge 2 \} $是一组合理的完美功率，让$ s \ subseteq \ mathscr {p} _ \ mathbb {q} $为有限的子集。我们证明了\ mathbb {z} [x] $的多项式$ f_s \的存在，因此$ f（\ mathbb {q}）\ cap \ mathscr {p} _ \ mathbb {q} = s $。这概括了Gajović的最新定理，他最近证明了类似的整数完美力量子集类似的定理。我们的方法利用了由于Ellenberg和其他人而导致的签名$（2,4，N）$的普遍的Fermat方程，以及Pethő和Shorey and Stewart证明的非分类二进制复发序列中完美幂的有限。
Let $\mathscr{P}_\mathbb{Q}=\{ α^n \; : \; α\in \mathbb{Q}, \; n \ge 2\}$ 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 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.
