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On Thue’s equation. (English) Zbl 0614.10018
In this very interesting paper the authors consider the number N(F,h) of solutions of Thue’s equation $$F(x,y)=h$$ in integers x,y$$\in {\mathbb{Z}}$$ with $$(x,y)=1$$. Here, $$F\in {\mathbb{Z}}[x,y]$$ is a binary form of degree $$r\geq 3$$ and h is fixed. In 1983 J. H. Evertse was the first to give a bound for N(F,h) depending only on r and h, but independent of the coefficients of F. The dependence on r is of the form $$\exp (r^ 2)$$. In the present paper the authors give a vast improvement of Evertse’s theorem. They show $$N(F,h)<c_ 1r^{1+t}$$, where t is the number of distinct prime divisors of h and $$c_ 1$$ is some computable constant which is $$<215$$ if r is big enough. In case $$h=1$$ this bound is optimal in its dependence on r, since $$x^ r+(y-x)(y-2x)...(y-rx)=1$$ obviously has at least r solutions.
The proof starts with the case $$h=1$$ and then, by induction on t, is extended to general h. To bound the number of large solutions the classical Thue-Siegel method is used. For the small solutions the authors first split the original problem in a union of Thue equations with large discriminant and then use some ingenious proximity arguments to finish the proof.
Reviewer: F.Beukers

##### MSC:
 11D41 Higher degree equations; Fermat’s equation
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##### References:
  Bombieri, E., Mueller, J.: On effective measures of irrationality for $$\sqrt[r]{{\frac{a}{b}}}$$ and related numbers. J. Reine Angew. Math.342, 173-196 (1983) · Zbl 0516.10024  Evertse, J.-H.: Upper bounds for the number of solutions of diophantine equations. Math. Centrum. Amsterdam, pp. 1-127 (1983) · Zbl 0517.10016  Evertse, J.-H.: On the representation of integers by binary cubic forms of positive discriminant. Invent. Math.73, 117-138 (1983) · Zbl 0506.10013 · doi:10.1007/BF01393828  Hooley, C.: On the representations of numbers by binary cubic forms. Glasgow Math. J.27, 95-98 (1985) · Zbl 0577.10021 · doi:10.1017/S0017089500006108  Lewis, D., Mahler, K.: Representation of integers by binary forms. Acta Arith.6, 333-363 (1961) · Zbl 0102.03601  Mahler, K.: An inequality for the discriminant of a polynomial. Michigan Math. J.11, 257-262 (1964) · Zbl 0135.01702 · doi:10.1307/mmj/1028999140  Mahler, K.: On Thue’s theorem. Austr. Nat. Univ. Res. Rep. 24 (1982). Math. Scand.55, 188-200 (1984) · Zbl 0544.10014  Siegel, C.L.: Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys. Math. Kl. (1929), N$$\deg$$ 1. Gesammelte Abhandlungen, Bd. 1, pp. 209-266. Berlin Heidelberg New York: Springer-Verlag 1966 · JFM 56.0180.05  Silverman, J.H.: Representation of integers by binary forms and the rank of the Mordell-Weil group. Invent. Math.74, 218-292 (1983) · Zbl 0525.14012 · doi:10.1007/BF01394317
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