论文标题
多项式的diophantine方程$ f（x）= g（y）$，具有简单的理性根部
The Diophantine equation $f(x)=g(y)$ for polynomials with simple rational roots
论文作者
论文摘要
在本文中，我们考虑了$ f（x）= g（y）$的diophantine方程，其中$ f $具有简单的理性根，$ g $具有有理系数。对于以有限的分母在理由中具有无限多种解决方案的情况下，我们给出了严格的条件。我们举例说明给定条件是必要的。事实证明，具有无限多种解决方案的这种方程与prouhet-tarry-escott元组密切相关。在特殊但重要的情况下，$ g $也只有简单的理性根源，我们可以提供更简单的声明。另外，我们还提供了属于限制长度连续整数块的术语的均等产品的应用。后者定理与Erdős和Turk的问题和结果以及Erdős和Graham有关。
In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where $f$ has simple rational roots and $g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples. In the special, but important case when $g$ has only simple rational roots as well, we can give a simpler statement. Also we provide an application to equal products with terms belonging to blocks of consecutive integers of bounded length. The latter theorem is related to problems and results of Erdős and Turk, and of Erdős and Graham.
