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Solutions entières de l’équation \(Y^ m=f(X)\). (Integer solutions of the equation \(Y^ m=f(X))\). (French) Zbl 0733.11009

Effective upper bounds are given for the size of the solutions in algebraic integers of a given number field \({\mathbb{K}}\) of equations \(Y^ m=f(X)\), where either \(m=2\) and \(f\in {\mathbb{K}}[X]\) has at least three zeros of odd order, or \(m\geq 3\) and \(f\in {\mathbb{K}}[X]\) has at least 2 zeros of order prime to m. The actual bounds are too involved to reproduce here.
A generalization of a method of W. M. Schmidt is used [cf. Integer points on curves of genus 1, Compos. Math. 81, No.1, 33-59 (1992)], and the results improve known estimates of B. Brindza [Acta Math. Hung. 44, 133-139 (1984; Zbl 0552.10009), and L. A. Trelina [Dokl. Akad. Nauk BSSR 22, 881-884 (1978; Zbl 0395.10026)].

MSC:

11D41 Higher degree equations; Fermat’s equation
11D75 Diophantine inequalities
11J25 Diophantine inequalities
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References:

[1] Baker, A., Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Phil. Soc.65 (1969), 439-444. · Zbl 0174.33803
[2] Brindza, B., On S-integral solutions of the equation ym = f(x)., Acta Math. Hung.44 (1984), 133-139. · Zbl 0552.10009
[3] Gyory, K., On the solutions of linear diophantine equations in Algebraic integers of bounded norm, Ann. Univ. Budapest Eotvos, Sect. Math.22-23 (1979-80), 225-233. · Zbl 0442.10010
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