Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T14:04:52.750Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds for the size of integral solutions to Ym = f(X)

Published online by Cambridge University Press:  20 January 2009

Dimitrios Poulakis
Dimitrios Poulakis
Affiliation:
Aristotle University of Thessaloniki, Department of Mathematics 54006 Thessaloniki, Greece, E-mail address: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be an algebraic number field with ring of integers OK and f(X) ∈ OK[X]. In this paper we establish improved explicit upper bounds for the size of solutions in OK, of diophantine equations Y2 = f(X), where f(X) has at least three roots of odd order, and Ym = f(X), where m is an integer ≥ 3 and f(X) has at least two roots of order prime to m.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 1 , February 1999 , pp. 127 - 141
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Baker, A., Bounds for the solutions of the hyperelliptic equation, Proc. Cambr. Philos. Soc. 65 (1969), 439–444.CrossRefGoogle Scholar
2.Kubert, D. and Lang, A., Units in modular function field I, Math. Ann. (1975), 67–96.CrossRefGoogle Scholar
3.Lang, S., Fundamentals of Diophantine Geometry, New York-Berlin-Heidelberg-Tokyo: Springer-Verlag, 1983.CrossRefGoogle Scholar
4.Lang, S., Introduction to algebraic and abelian Functions, New York-Berlin-Heidelberg: Springer-Verlag, 1982.CrossRefGoogle Scholar
5.Poulakis, D., Solutions entiéres de l'équation f(X, Y)x = h(X)g(X, Y), C.R. Acad. Sci. Paris 315 (1992), 963–968.Google Scholar
6.Poulakis, D., Integer points on algebraic curves with exceptional units, J. Austral. Math. Soc. (Series A) 63 (1997), 145–164.CrossRefGoogle Scholar
7.Schmidt, W., Eisenstein–s theorem on power series expansions of algebraic functions, Acta Arith. LVI (1990), 161–179.CrossRefGoogle Scholar
8.Serre, J. P., Corps Locaux, Hermann, Paris, 1962.Google Scholar
9.Silverman, J. H., The Arithmetic of elliptic curves, New York-Berlin-Heidelberg: Springer-Verlag, 1986.CrossRefGoogle Scholar
10.Voutier, P., An Upper Bound for the Size of Integral Solutions to Ym = f(X), J. Number Theory 53 (1995), 247–271.CrossRefGoogle Scholar