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Bounds for the solutions of some diophantine equations in terms of discriminants

Part of: Diophantine equations

Published online by Cambridge University Press:  09 April 2009

B. Brindza ,
J. H. Evertse  and
K. Györy
B. Brindza
Affiliation:
Mathematical Institute Kossuth Lajos University4010 Debrecen Hungary
J. H. Evertse
Affiliation:
Department of Mathematics and Computer Science 2300 RA Leiden P.O. Box 9512 The Netherlands
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Abstract

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Several effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation 11D61: Exponential equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 1 , August 1991 , pp. 8 - 26
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Baker, A., ‘Contributions to the theory of diophantine equations’, Philos. Trans. Roy. Soc. London A 263 (1968), 173–208.Google Scholar
[2]Baker, A., ‘Bounds for the solutions of the hyperelliptic equation’, Proc. Cambridge Philos. Soc. 65 (1969), 439–444.CrossRefGoogle Scholar
[3]Brindza, B., ‘Thue equations and multiplicative independence’, Number theory and cryptography (Loxton, J. H., ed.), Cambridge Univ. Press, 1990, pp. 213–220.Google Scholar
[4]Coates, J., ‘An effective p-adic analogue of a theorem of Thue’, Acta Arith. 15 (1969), 279–305.CrossRefGoogle Scholar
[5]Evertse, J. H. and Győry, K., ‘Decomposable form equations’, New Advances in Transcendence Theory (Baker, A., ed.), Cambridge Univ. Press, 1988, pp. 175–202.Google Scholar
[6]Evertse, J. H. and Győry, K., ‘Effective finiteness results for binary forms with given discriminant’, to appear.Google Scholar
[7]Evertse, J. H., Győry, K., Stewart, C. L. and Tijdeman, R., ‘On S-unit equations’, Invent. Math. 92 (1988), 461–477.Google Scholar
[8]Győry, K., ‘Sur les polynômes à coefficients entiers et de discriminant donné II.’, Publ. Math. Debrecen 21 (1974), 125–144.Google Scholar
[9]Győry, K., ‘On polynomials with integer coefficients and given discriminant IV.’, Publ. Math. Debrecen 25 (1978), 155–167.Google Scholar
[10]Győry, K., ‘On the number of solutions of linear equations in units of an algebraic number field’, Comment. Math. Helv. 54 (1979), 583–600.Google Scholar
[11]Győry, K., ‘On the solutions of linear diophantine equations in algebraic integers of bounded norm’, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 22/23 (1980), 225–233.Google Scholar
[12]Győry, K., ‘Explicit upper bounds for the solutions of some diophantine equations’, Ann. Acad. Sci. Fenn. Ser. A 15 (1980), 3–12.Google Scholar
[13]Győry, K., ‘Résultats effectifs sur la représentation des entiers par des formes décomposables’, Queen's Papers in Pure and Applied Math., No. 56, Kingston, Canada, 1980.Google Scholar
[14]Győry, K., ‘On S-integral solutions of norm form, discriminant form and index form equations’, Studia Sci. Math Hungar. 16 (1981), 149–161.Google Scholar
[15]Lang, S., ‘Fundamentals of diophantine geometry’, Springer-Verlag, 1983.CrossRefGoogle Scholar
[16]Philippon, P. and Waldschmidt, M., ‘Lower bounds for linear forms in logarithms’, New Advances in Transcendence Theory (Baker, A., ed.), Cambridge Univ. Press, 1988, pp. 280–312.CrossRefGoogle Scholar
[17]Rosser, J. B. and Schoenfeld, L., ‘Approximate formulas for some functions of prime numbers’, Illinois J. Math. 6 (1962), 64–94.CrossRefGoogle Scholar
[18]Schinzel, A. and Tijdeman, R., ‘On the equation y m = P(x)’, Acta Arith. 31 (1976), 199–204.Google Scholar
[19]Shorey, T. N., van der Poorten, A. J., Tijdeman, R. and Schinzel, A., ‘Applications of the Gel'fond-Baker method to diophantine equations’, Transcendence Theory: Advances and Applications (Baker, A. and Masser, D. W., eds.), Academic Press, London, 1977, pp. 59–77.Google Scholar
[20]Shorey, T. N. and Tijdeman, R., Exponential diophantine equations, Cambridge Univ. Press, 1986.CrossRefGoogle Scholar
[21]Siegel, C. L., ‘Abschätzung von Einheiten’, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 2 (1969), 71–86.Google Scholar
[22]Sprindžuk, V. G., Classical diophantine equations in two unknowns, Nauka, Moskva, 1982. (Russian)Google Scholar
[23]Stark, H. M., ‘Some effective cases of the Brauer-Siegel theorem’, Invent. Math. 23 (1974), 135–152.CrossRefGoogle Scholar
[24]Trelina, L. A., ‘Representations of powers by polynomials in algebraic number fields’, Dokl. Akad. Nauk BSSR 29 (1985), 5–8, in Russian.Google Scholar
[25]Zimmert, R., ‘Ideale kleiner Norm in Idealidassen und eine Regulatorabschatzung’, Invent. Math. 62 (1981), 367–380.CrossRefGoogle Scholar