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Counting solutions to trinomial Thue equations: a different approach

Author: Emery Thomas
Journal: Trans. Amer. Math. Soc. 352 (2000), 3595-3622
MSC (2000): Primary 11D41, 11J68; Secondary 11Y50
Published electronically: March 16, 2000
MathSciNet review: 1641119
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Abstract | References | Similar Articles | Additional Information


We consider the problem of counting solutions to a trinomial Thue equation -- that is, an equation

\begin{equation*}\vert F(x,y)\vert = 1,\tag{$*$ } \end{equation*}

where $F$ is an irreducible form in $Z[x,y]$ with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the ``Thue-Siegel principle" and its relation to $(*)$. In this paper we give specific numerical bounds for the number of solutions to $(*)$ by a somewhat different approach, the difference lying in the initial step -- solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.

References [Enhancements On Off] (What's this?)

  • [1] Baker, A. (1964). Rational approximations to certain algebraic numbers. Proc. London Math Soc. (3) 14, 385-398. MR 28:5029
  • [2] Baker, A. (1968). Contributions to the theory of diophantine equations. Phil. Trans. Roy. Soc. London A263 173-208. MR 37:4005; MR 37:4006
  • [3] Bombieri, E. (1982). On the Thue-Siegel-Dyson Theorem. Acta Math. 148, 255-296. MR 83m:10052
  • [4] Bombieri, E. and Mueller, J. (1983). On effective measures of irrationality for $\sqrt[r]{(a/b)}$ and related numbers. J. Reine Angew. Math. 342, 173-196. MR 84m:10023
  • [5] Bombieri, E. and Schmidt, W. M. (1987). On Thue's equation, Invent. Math. 88, 69-81. MR 88d:11026
  • [6] Davenport, H. and Roth, K. F. (1955). Rational approximations to algebraic numbers. Mathematika 3, 160-167. MR 17:1060d
  • [7] Dyson, F. J. (1947). The approximation to algebraic numbers by rationals. Acta Math. 79, 225-240. MR 9:412h
  • [8] Evertse, J. H. (1983). Upper bounds for the numbers of solutions of Diophantine equations. Math Centrum tract 168, Amsterdam.MR 85k:11015
  • [9] Feldman, N. I. (1968). Estimation of a linear form of logarithms of algebraic numbers, Mat. Sb. 77 (119), 423-436; English transl., Math. USSR Sb. 6 (1968), 393-406. MR 38:1059
  • [10] Gelfond, A. O. (1952). Transcendental and Algebraic Numbers, Dover, New York, 1960. MR 22:2598
  • [11] Lagrange, J. L. (1867). Oeuvres, vol. 2, section 2, Gauthier-Villars, Paris, 675-693. MR 55:12437 (reprint)
  • [12] Lewis, D. and Mahler, K. (1961). On the representation of integers by binary forms. Acta Arith. 6, 333-363. MR 22:10952
  • [13] Mahler, K. (1984). On Thue's Theorem, Math. Scand. 55, 188-200. MR 86h:11030
  • [14] Mignotte, M. (1992). Mathematics for Computer Algebra, Springer-Verlag, New York. MR 92i:68071
  • [15] Mueller, J. (1987). Counting solutions of $\vert ax^r-by^r\vert\leq h$. Quarterly J. Math. Oxford Ser. (2) 32, 503-513. MR 89b:11026
  • [16] Mueller, J. and Schmidt, W. M. (1987). Trinomial Thue equations and inequalities. J. Reine Angew. Math. 379, 76-99.MR 88j:11011
  • [17] Mueller, J. and Schmidt, W. M. (1988). Thue's equation and a conjecture of Siegel. Acta Math. 160, 207-247. MR 89g:11029
  • [18] Roth, K. F. (1955). Rational approximations to algebraic numbers, Mathematika 2, 1-20. MR 17:242d
  • [19] Schmidt, W. M. (1991). Diophantine Approximations and Diophantine Equations, LNM 1467, Springer-Verlag, New York. MR 94f:11059
  • [20] Schneider, T. (1957). Einfuhrung in die transzendenten Zahlen, Springer-Verlag, Berlin. MR 19:252f
  • [21] Siegel, C. L. (1929). Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss., Math. Phys. Kl., Nr. 1=Ges. Abh. I., 209-266.
  • [22] Stewart, C. L. (1991). On the number of solutions of polynomial congruences and Thue equations. J. Amer. Math. Soc. 4, 793-835.MR 92j:11032
  • [23] Thue, A. (1909). Über Annäherungswerte algebraischer Zahlen. J. Reine Angew. Math. 135, 284-305.

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Additional Information

Emery Thomas
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720–3840

Keywords: Thue equation, Thue-Siegel principle, diophantine approximation, trinomial equation, counting solutions.
Received by editor(s): May 23, 1997
Received by editor(s) in revised form: July 29, 1998
Published electronically: March 16, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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