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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:
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