Thue equations with few coefficients
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- by Wolfgang M. Schmidt PDF
- Trans. Amer. Math. Soc. 303 (1987), 241-255 Request permission
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 $|F(x, y)| \leqslant h$ has $\ll {(rs)^{1/2}}{h^{2/r}}(1 + \log {h^{1/r}})$ solutions.References
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E. Bombieri and W. M. Schmidt, On Thue’s equation, Invent. Math. (to appear).
- K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257–262. MR 166188 J. Mueller, Counting solutions of $|a{x^r} - b{y^r}| \leqslant h$, Quart. J. Math. Oxford (to appear). J. Mueller and W. M. Schmidt, The number of solutions of trinomial Thue equations and inequalities, Crelle’s J. (to appear). —, Thue equations and a conjecture of Siegel, Acta Math. (submitted).
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 303 (1987), 241-255
- MSC: Primary 11D41; Secondary 11D75
- DOI: https://doi.org/10.1090/S0002-9947-1987-0896020-1
- MathSciNet review: 896020