Polynomial values of products of terms from an arithmetic progression
In [Acta Arith. 95, No. 3, 261--288 (2000; Zbl 0958.11049)] \textit{Y. F. Bilu} and \textit{R. F. Tichy} proved a remarkable result that enables a useful way to apply Siegel's theorem on integral points which is not effective, to prove finiteness theorems on Diophantine equations. Since then, there has been a deluge of papers on Diophantine equations of the form \(f(x)=g(y)\) for polynomials \(f,g\) to which the first author has also contributed extensively. In the paper under review, the authors consider the Diophantine equations of the form \(f_{k,j}(x) = g(y)\), where \(g \in \mathbb{Q}[y]\) has degree \(\geq 2\) and \[f_{k,j}(x) = x(x+d) \cdots (x+(j-1)d)(x+(j+1)d) \cdots (x+(k-1)d)\] where \(d \neq 0, 0 \leq j < k\). Note that one takes \(k \geq 3\) so that we do have a polynomial. Using the Bilu-Tichy theorem, they obtain finiteness results when \(k \geq 8\); the results are ineffective of course. In the special case, when \(g(y) = ay^n+b\) with \(n \geq 2\) and \(a,b \in \mathbb{Q}\), they are able to obtain effective results. Finally, the authors obtain (in the reviewer's opinion, the most interesting) results on the decomposability of the polynomials \(f_{j,k}(x)\).
scientific article; zbMATH DE number 7259090
scientific article