## 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

### Citations:

Zbl 0552.10009; Zbl 0395.10026
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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 [4] Lang, S., Fundamentals of Diophantine Geometry, Springer-Verlag (1983). · Zbl 0528.14013 [5] Lang, S., Algebraic Number Theory, Addison Wesley (1970). · Zbl 0211.38404 [6] Schmidt, W., Integer points on curves of genus1 (à paraître). [7] Siegel, C.L., Abschätzung von Einheiten, Nachr. Akd.Wiss Göttingen Math. Phys. K1II (1969), 71-86. · Zbl 0186.36703 [8] Sprindzuk, V.G., A hyperelliptic diophantine equation and class numbers (in Russian), Acta Arith.30 (1976), 95-108. · Zbl 0335.10021 [9] Trelina, L.A., On S-integral solutions of the hyperelliptic equation (in Russian), Dokl.Akad. Nauk BSSR (1978), 881-884. · Zbl 0395.10026
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