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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Thue equations with few coefficients

Author: Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 303 (1987), 241-255
MSC: Primary 11D41; Secondary 11D75
MathSciNet review: 896020
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Abstract: Let $ F(x,\,y)$ be a binary form of degree $ r \geqslant 3$ with integer coefficients, and irreducible over the rationals. Suppose that only $ s + 1$ of the $ r + 1$ coefficients of $ F$ are nonzero. Then the Thue equations $ F(x,\,y) = 1$ has $ \ll {(rs)^{1/2}}$ solutions. More generally, the inequality $ \vert F(x,\,y)\vert \leqslant h$ has $ \ll {(rs)^{1/2}}{h^{2/r}}(1 + \log {h^{1/r}})$ solutions.

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

  • [1] E. Bombieri and W. M. Schmidt, On Thue's equation, Invent. Math. (to appear).
  • [2] K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257–262. MR 166188
  • [3] J. Mueller, Counting solutions of $ \vert a{x^r} - b{y^r}\vert \leqslant h$, Quart. J. Math. Oxford (to appear).
  • [4] J. Mueller and W. M. Schmidt, The number of solutions of trinomial Thue equations and inequalities, Crelle's J. (to appear).
  • [5] -, Thue equations and a conjecture of Siegel, Acta Math. (submitted).

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Article copyright: © Copyright 1987 American Mathematical Society