On Thue’s equation. (English) Zbl 0614.10018

Bombieri, E.; Schmidt, W. M.

Invent. Math. 88, 69-81 (1987).

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.

Reviewer: Frits Beukers (Utrecht)

MSC:

11D45	Counting solutions of Diophantine equations
11D59	Thue-Mahler equations

Keywords:

upper bounds; primitive solutions; Thue equation; binary form of higher degree; large solutions; small solutions

References:

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