On the equation \(f(x)=y^ m\) over finitely generated domains. (English) Zbl 0691.10006
The purpose of the paper is to give the first effective upper bounds for the solutions of hyperelliptic equations considered over integral domains, finitely generated over \(\mathbb Z\). This type of equations plays an essential role in the theory and applications of diophantine equations.
Ineffective theorems on hyperelliptic equations over number fields were obtained by C. L. Siegel and W. J. LeVeque. These results were extended to the case of integral domains of finite type over \(\mathbb Z\) by S. Lang. The first effective bounds for the solutions of hyperelliptic equations over number fields were given by A. Baker, whose result was extended and improved by several authors. In the function field case W. M. Schmidt, R. C. Mason and the author obtained effective theorems.
In case of integral domains, finitely generated over \(\mathbb Z\), hyperelliptic equations can not be reduced to Thue equations directly. Hence the proof of the main theorem combines the effective result of R. C. Mason and the author [Acta Arith. 47, No. 2, 167–173 (1986; Zbl 0561.10009)] on the solutions of hyperelliptic equations over function fields with the result of the author [Acta Math. Hung. 44, No. 1–2, 133–139 (1984; Zbl 0552.10009)] on S-integral solutions of hyperelliptic equations over number fields by applying the technics developed by K. Györy [Acta Math. Hung. 42, No. 1–2, 45–80 (1983; Zbl 0528.10014)].
Ineffective theorems on hyperelliptic equations over number fields were obtained by C. L. Siegel and W. J. LeVeque. These results were extended to the case of integral domains of finite type over \(\mathbb Z\) by S. Lang. The first effective bounds for the solutions of hyperelliptic equations over number fields were given by A. Baker, whose result was extended and improved by several authors. In the function field case W. M. Schmidt, R. C. Mason and the author obtained effective theorems.
In case of integral domains, finitely generated over \(\mathbb Z\), hyperelliptic equations can not be reduced to Thue equations directly. Hence the proof of the main theorem combines the effective result of R. C. Mason and the author [Acta Arith. 47, No. 2, 167–173 (1986; Zbl 0561.10009)] on the solutions of hyperelliptic equations over function fields with the result of the author [Acta Math. Hung. 44, No. 1–2, 133–139 (1984; Zbl 0552.10009)] on S-integral solutions of hyperelliptic equations over number fields by applying the technics developed by K. Györy [Acta Math. Hung. 42, No. 1–2, 45–80 (1983; Zbl 0528.10014)].
Reviewer: István Gaál (Debrecen)
MSC:
| 11D41 | Higher degree equations; Fermat’s equation |
References:
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