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

Abstract

Let $\ell$ and $p$ be (not necessarily distinct) prime numbers and $F$ be a global function field of characteristic $\ell$ with field of constants $\kappa$. 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 $\kappa$. 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.