论文标题
$ y^n = f（x）$上方的diophantine方程
Diophantine equations of the form $Y^n=f(X)$ over function fields
论文作者
论文摘要
令$ \ ell $和$ p $为（不一定是不同的）质数，而$ f $是具有常数$κ$字段的特征$ \ ell $的全球功能字段。假设存在具有$ 1 $的prime $ p_ \ infty $的$ f $，然后让$ \ natercal {o} _f $是$ f $的子元素，由$ f $组成，没有杆子，远离$ p_ \ infty $。令$ f（x）$为$ x $的多项式，系数为$κ$。我们研究了$ y^{n} = f（x）$的diophantine方程的解决方案，该方程为$ \ mathcal {o} _f $，尤其是表明，如果$ m $ $ f（x）$满足其他条件，则没有非构成解决方案。获得的结果适用于在$ \ Mathbb {z} _ {p} $ - $ f $的扩展中，在某些整数中对$ y^{n} = f（x）$的解决方案的研究。我们证明了在多项式环$ k [t_1，\ dots，t_r] $中的解决方案的结果，其中$ k $是任何特征$ \ ell $的字段，表明唯一的解决方案必须在$ k $中。我们将我们的方法应用方法来研究$ y^n = \ sum_ {i = 1}^d（x+ir）^m $的diophantine方程的解决方案，其中$ m，n，d \ geq 2 $是整数。
Let $\ell$ and $p$ be (not necessarily distinct) prime numbers and $F$ be a global function field of characteristic $\ell$ with field of constants $κ$. Assume that there exists a prime $P_\infty$ of $F$ which has degree $1$, and let $\mathcal{O}_F$ be the subring of $F$ consisting of functions with no poles away from $P_\infty$. Let $f(X)$ be a polynomial in $X$ with coefficients in $κ$. We study solutions to diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal{O}_F$, and in particular, show that if $m$ and $f(X)$ satisfy additional conditions, then there are no non-constant solutions. The results obtained apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb{Z}_{p}$-extensions of $F$ known as constant $\mathbb{Z}_p$-extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \dots, T_r]$, where $K$ is any field of characteristic $\ell$, showing that the only solutions must lie in $K$. We apply our methods to study solutions of diophantine equations of the form $Y^n=\sum_{i=1}^d (X+ir)^m$, where $m,n, d\geq 2$ are integers.
