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Poulakis, Dimitrios

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

Sémin. Théor. Nombres Bordx., Sér. II 3, No. 1, 187-199 (1991).
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)].
Reviewer: R.J.Stroeker (Rotterdam)

Cited in 5 Documents

MSC:

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

Keywords:

Effective upper bounds; size of the solutions in algebraic integers

Citations:

Zbl 0552.10009; Zbl 0395.10026
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\textit{D. Poulakis}, Sémin. Théor. Nombres Bordx., Sér. II 3, No. 1, 187--199 (1991; Zbl 0733.11009)
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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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