On large values of binary forms
Let \(F(X,Y)\) be an irreducible binary form of degree \(n\geq 3\) with coefficients in \({\mathbb{Z}}\), and let \(m\) be a positive integer having \(s\) distinct prime factors. The author deals with the Thue equation (*) \(|F(x,y)|=m\) in coprime integers \(x,y\), where solutions \((x,y)\) and \((-x,-y)\) are considered equal. \textit{E. Bombieri} and \textit{W. M. Schmidt} [Invent. Math. 88, 69-81 (1987; Zbl 0614.10018)] showed that (*) has at most \(cn^{1+s}\) solutions, where \(c\) is an absolute constant which for \(n\) sufficiently large can be taken equal to \(215\). The author shows that if \(n\geq 6\) and \(m\) is larger than an effectively computable number \(C\), depending only on the discriminant of \(F\), then (*) has at most \(n\big((n^{4/(n-2)})^s +6\big)\) solutions. In his proof, the author estimates the number of solutions which are ``large'' in terms of \(m\) in the usual way, using the Thue-Siegel principle and a gap principle. The author treats the ``small'' solutions by means of a new idea, which uses a theorem of \textit{K. Györy} and \textit{J.-H. Evertse} [Compos. Math. 79, 169-204 (1991; Zbl 0746.11020)], stating that every irreducible binary form \(F\in {\mathbb{Z}}[X,Y]\) of discriminant \(D\) is SL\(_2({\mathbb{Z}})\)-equivalent to a binary form \(G\) whose coefficients are effectively bounded in terms of \(D\).
scientific article; zbMATH DE number 1026391
scientific article