## 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 $$r\ge 3$$ 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 $$x^r+(y-x)(y-2x) \cdots (y-rx)=1$$ 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.

### MSC:

 11D45 Counting solutions of Diophantine equations 11D59 Thue-Mahler equations
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### References:

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