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DIOPHANTINE EQUATIONS OF THE FORM $Y^n=f(X)$ OVER FUNCTION FIELDS.
- Source :
-
Bulletin of the Australian Mathematical Society . Dec2023, Vol. 108 Issue 3, p379-390. 12p. - Publication Year :
- 2023
-
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 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00049727
- Volume :
- 108
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Bulletin of the Australian Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 173490762
- Full Text :
- https://doi.org/10.1017/S0004972723000412