论文标题
关于功能字段的加泰罗尼亚方程的评论
Remarks on Catalan's equation over function fields
论文作者
论文摘要
令$ \ ell $为质数，$ f $是特征$ \ ell $的全局功能字段。假设有一个Prime $ p_ \ infty $ of度数$ 1 $。令$ \ mathcal {o} _f $为$ f $中的函数环，没有杆子以外的$ \ {p_ \ infty \} $。我们研究了加泰罗尼亚方程的解决方案$ x^m-y^n = 1 $ over $ \ mathcal {o} _f $，并表明在某些其他条件下，没有$ \ Mathcal {o} _f $，当$ m，m，n> 1 $时，没有非恒定解决方案。
Let $\ell$ be a prime number, $F$ be a global function field of characteristic $\ell$. Assume that there is a prime $P_\infty$ of degree $1$. Let $\mathcal{O}_F$ be the ring of functions in $F$ with no poles outside of $\{P_\infty\}$. We study solutions to Catalan's equation $X^m-Y^n=1$ over $\mathcal{O}_F$ and show that under certain additional conditions, there are no non-constant solutions which lie in $\mathcal{O}_F$, when $m,n>1$.
