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Transactions of the American Mathematical Society

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Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge's theorem

Authors: David Lee Hilliker and E. G. Straus
Journal: Trans. Amer. Math. Soc. 280 (1983), 637-657
MSC: Primary 11D41
MathSciNet review: 716842
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Abstract: In 1887 Runge [13] proved that a binary Diophantine equation $ F(x,y) = 0$, with $ F$ irreducible, in a class including those in which the leading form of $ F$ is not a constant multiple of a power of an irreducible polynomial, has only a finite number of solutions. It follows from Runge's method of proof that there exists a computable upper bound for the absolute value of each of the integer solutions $ x$ and $ y$. Runge did not give such a computation. Here we first deduce Runge's Theorem from a more general theorem on Puiseux series that may be of interest in its own right. Second, we extend the Puiseux series theorem and deduce from the generalized version a generalized form of Runge's Theorem in which the solutions $ x$ and $ y$ of the polynomial equation $ F(x,y) = 0$ are integers, satisfying certain conditions, of an arbitrary algebraic number field. Third, we compute bounds for the solutions $ (x,y) \in {{\mathbf{Z}}^2}$ in terms of the height of $ F$ and the degrees in $ x$ and $ y$ of $ F$.

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  • [1] William J. Ellison, Variations sur un thème de Carl Runge, Séminaire Delange-Pisot-Poitou (13e année:1971/72), Théorie des nombres, Fasc. 1, Exp. No. 9, Secrétariat Mathématique, Paris, 1973, pp. 4 (French). MR 0419348
  • [2] E. Heine, Handbuch der Kugelfunctionen. Theorie und Anwendungen. Vols. I, II, 2nd ed., G. Reimer, Berlin, 1878; 1881. See Jbuch. 10, 332; 13, 390-391.
  • [3] David Lee Hilliker, An algorithm for solving a certain class of Diophantine equations. I, Math. Comp. 38 (1982), no. 158, 611–626. MR 645676,
  • [4] -, An algorithm for solving a certain class of Diophantine equations. II (to be submitted).
  • [5] -, An algorithm for computing the values of the ramification index in the Puiseux series expansions of an algebraic function (to be submitted).
  • [6] David Lee Hilliker and E. G. Straus, On Puiseux series whose curves pass through an infinity of algebraic lattice points, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 59–62. MR 682822,
  • [7] William Judson LeVeque, Topics in number theory. Vols. 1 and 2, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682
  • [8] Edmond Maillet, Sur les équations indéterminées à deux et trois variables qui n'ont qu'un nombre fini de solutions en nombres entiers, J. Math. Pures Appl. 6 (1900), 261-277. An abstract appeared in C. R. Acad. Sci. Paris 128 (1899), 1383-1395. See Jbuch. 30, 188-189; 31, 190-191.
  • [9] -, Sur une catégorie de'équations indéterminées n'ayant en nombres entiers qu'un nombre fini de solutions, Nouv. Ann. Math. 18 (1918), 281-292. See Jbuch. 46, 210.
  • [10] L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
  • [11] Harry Pollard, The Theory of Algebraic Numbers, Carus Monograph Series, no. 9, The Mathematical Association of America, Buffalo, N. Y., 1950. MR 0037319
  • [12] G. Pólya and G. Szegö, Problems and theorems in analysis, Vols. I, II, Revised and enlarged transl. of 4th German ed., Die Grundlehren der Math. Wissenschaften, Bands 193, 216, Springer-Verlag, New York and Berlin, 1972, 1976; 1st German ed., Aufgaben und Lehrsätze aus der Analysis, Julius Springer, Berlin, 1925; 4th German ed., 1970, 1971; 1st German ed. also published in two volumes by Dover, New York, 1945.
  • [13] C. Runge, Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen, J. Reine Angew. Math. 100 (1887), 425-435. See Jbuch. 19, 76-77.
  • [14] A. Schinzel, An improvement of Runge’s theorem on Diophantine equations, Comment. Pontificia Acad. Sci. 2 (1969), no. 20, 1–9 (English, with Latin summary). MR 0276174
  • [15] Carl Siegel, Approximation algebraischer Zahlen, Math. Z. 10 (1921), no. 3-4, 173–213 (German). MR 1544471,
  • [16] -, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Natur. K1. 1 (1929). Also in Gesammelte Abhandlungen Vol. I, Springer-Verlag, Berlin and New York. 1966, pp. 209-266. See Jbuch. 56, 180-184.
  • [17] Th. Skolem. Über ganzzahlige Löhungen einer Klasse unbestimmter Gleichungen, Norsk mat. Foren. Akrifter, Ser. I 10 (1922). See Jbuch. 48, 139.
  • [18] -, Diophantische Gleichungen, Verlag von Julius Springer, Berlin, 1938, reprinted by Chelsea, New York, 1950.

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Keywords: Algebraic function, Diophantine equation, Puiseux series
Article copyright: © Copyright 1983 American Mathematical Society