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On the resolution of inhomogeneous norm form equations in two dominating variables


Author: István Gaál
Journal: Math. Comp. 51 (1988), 359-373
MSC: Primary 11D57
DOI: https://doi.org/10.1090/S0025-5718-1988-0942162-6
MathSciNet review: 942162
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Abstract | References | Similar Articles | Additional Information

Abstract: Applying Baker's well-known method and the reduction procedure described by Baker and Davenport, we give a numerical algorithm for finding all solutions of inhomogeneous Thue equations of type

$\displaystyle {N_{K/Q}}(x + \alpha y + \lambda ) = 1$

in the variables $ x,y \in Z$ and $ \lambda \in {Z_K}$ with $ \lceil \lambda \rceil < (\max \vert x\vert,\vert y\vert){)^{1/2}}$, where $ K = Q(\alpha )$ is a totally real cubic field.

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  • [1] A. Baker, "Contributions to the theory of Diophantine equations," Philos. Trans. Roy. Soc. London Ser. A, v. 263, 1968, pp. 173-208. MR 0228424 (37:4005)
  • [2] A. Baker, Transcendental Number Theory, 2nd ed., Cambridge Univ. Press, New York, 1979. MR 1074572 (91f:11049)
  • [3] A. Baker & H. Davenport, "The equations $ 3{x^2} - 2 = {y^2}$ and $ 8{x^2} - 7 = {z^2}$," Quart. J. Math. Oxford, v. 20, 1969, pp. 129-137. MR 0248079 (40:1333)
  • [4] J. Coates, "An effective p-adic analogue of a theorem of Thue," Acta Arith., v. 15, 1969, pp. 279-305. MR 0242768 (39:4095)
  • [5] W. J. Ellison, "Recipes for solving Diophantine problems by Baker's method," Séminaire de Théorie des Nombres, 1970-1971, Exp. No. 11, Lab. Théorie des Nombres, C.N.R.S., Talence, 1971. MR 0392880 (52:13693)
  • [6] W. J. Ellison, F. Ellison, J. Pesek, C. E. Stahl & D. S. Stall, "The Diophantine equation $ {y^2} + k = {x^3}$," J. Number Theory, v. 4, 1972, pp. 107-117. MR 0316376 (47:4923)
  • [7] I. Gaál, "Norm form equations with several dominating variables and explicit lower bounds for inhomogeneous linear forms with algebraic coefficients," Studia Sci. Math. Hungar., v. 19, 1984, pp. 399-411. MR 874508 (88e:11057a)
  • [8] I. Gaál, "Norm form equations with several dominating variables and explicit lower bounds for inhomogeneous linear forms with algebraic coefficients II," Studia Sci. Math. Hungar., v. 20, 1985, pp. 333-344. MR 886038 (88e:11057b)
  • [9] K. Györy, "Sur certaines généralisations de l'équation de Thue-Mahler," Enseign. Math., v. 26, 1980, pp. 247-255. MR 610525 (82e:10032)
  • [10] K. Györy, Résultats Effectifs sur la Représentation des Entiers par des Formes Décomposables, Queen's Papers in Pure and Appl. Math., No. 56, Kingston, Ontario, Canada, 1980.
  • [11] K. Györy, "On S-integral solutions of norm form, discriminant form and index form equations," Studia Sci. Math. Hungar., v. 16, 1981, pp. 149-161. MR 703653 (84g:10037)
  • [12] K. Györy, "Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains," Acta Math. Hungar., v. 42, 1983, pp. 45-80. MR 716553 (85f:11020)
  • [13] K. Györy & Z. Z. Papp, Norm Form Equations and Explicit Lower Bounds for Linear Forms with Algebraic Coefficients, Studies in Pure Math. (To the memory of Paul Turán), Akadémiai Kiadò, Budapest, 1983, pp. 245-257. MR 820227 (87b:11025)
  • [14] D. E. Knuth, The Art of Computer Programming, II, Addison-Wesley, Reading, Mass., 1966. MR 0378456 (51:14624)
  • [15] S. V. Kotov, On Diophantine Equations of Norm Form Type II, Inst. Mat. Akad. Nauk BSSR, Preprint No. 10, Minsk, 1980. (Russian)
  • [16] S. V. Kotov, Effective Bounds for Linear Forms with Algebraic Coefficients in Archimedean and p-Adic Metrics, Inst. Mat. Akad. Nauk BSSR, Preprint No. 24, Minsk, 1981. (Russian)
  • [17] S. V. Kotov, "Effective bound for the values of the solutions of a class of Diophantine equations of norm form type," Mat. Zametki, v. 33, 1983, pp. 801-806. (Russian) MR 709218 (84j:10014)
  • [18] A. Pethö, "On the resolution of Thue inequalities," J. Symb. Comput., v. 4, 1987, pp. 103-109. MR 908418 (89b:11030)
  • [19] A. Pethö & R. Schulenberg, Effektives Lösen von Thue Gleichungen, Publ. Math. Debrecen, v. 34, 1987, pp. 189-196. MR 934900 (89c:11044)
  • [20] T. N. Shorey & R. Tijdeman, Exponential Diophantine Equations, Cambridge Univ. Press, Oxford, 1986. MR 891406 (88h:11002)
  • [21] V. G. Sprindžuk, "Representation of numbers by the norm forms with two dominating variables," J. Number Theory, v. 6, 1974, pp. 481-486. MR 0354567 (50:7045)
  • [22] R. P. Steiner, "On Mordell's equation $ {y^2} - k = {x^3}$: A problem of Stolarsky," Math. Comp., v. 46, 1986, pp. 703-714. MR 829640 (87e:11041)
  • [23] R. Tijdeman, On the Gel'fond-Baker Method and Its Application, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R.I., 1976, pp. 241-268. MR 0434974 (55:7936)
  • [24] M. Waldschmidt, "A lower bound for linear forms in logarithms," Acta Arith., v. 37, 1980, pp. 257-283. MR 598881 (82h:10049)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1988-0942162-6
Keywords: Computer solution of Diophantine equations, Thue equation, Davenport's lemma
Article copyright: © Copyright 1988 American Mathematical Society

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