arXiv 2207.030802022-07-09math.NT

函数域上$Y^n=f(X)$型丢番图方程

Diophantine equations of the form $Y^n=f(X)$ over function fields

Anwesh Ray

备注：: 10 pages, some corrections and reformulations

摘要：

设$\ell$和$p$是（不一定是不同的）素数，$F$是特征$\ell$的全局函数域，其域为常数$κ$。设$F$的一个素数$P_\infty$的度为$1$，$\mathcal{O}_F$是$F$的一个子环，该子环由没有极点远离$P_\infty$的函数组成.令$f(X)$是$X$中的多项式，系数在$κ$中。我们研究了位于$\mathcal{O}_F$中的$Y^{n}=f(X)$形式的丢番图方程的解，特别地，证明了如果$m$和$f(X)$满足附加条件，则不存在非常数解。所得到的结果适用于研究的解决方案PDMATHAMBMV在某些环的整数在$\mathbb{Z}_{p}$-扩展的$F$称为常数$\mathbb{Z}_p$ -扩展。我们证明了类似的结果的解决方案在多项式环$K[T_1, \dots, T_r]$，其中$K$是任何领域的特征$\ell$，表明唯一的解决方案必须位于$K$。我们应用我们的方法来研究解丢番图方程的形式$Y^n=\sum_{i=1}^d (X+ir)^m$，其中$m,n, d\geq 2$是整数。

英文摘要：

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.

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