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DIOPHANTINE EQUATIONS OF THE FORM
$Y^n=f(X)$ OVER FUNCTION FIELDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 3 / December 2023
- Published online by Cambridge University Press:
- 18 May 2023, pp. 379-390
- Print publication:
- December 2023
-
- Article
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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 nonconstant solutions. The results 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 
$K[T_1, \ldots , T_r]$, 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.
