On a conjecture of Siegel
Let \(k\) be an algebraically closed field of characteristic 0, \(R=k[t]\). Let \(r\geq 2\) and \(a_1,\ldots,a_r,c\in R\) and consider the equation \[ \sum_{i=1}^r a_i x^{\alpha_{1i}} y^{\alpha_{2i}}=c \] in the variables \(x,y\in R\) subject to the conditions: (A) \(a_i\neq 0\) for \(i=1,\ldots,r\), (B) \(| \alpha_{\nu i}-\alpha_{\nu j}| \geq K\) for \(i\neq j\), (C) \(\alpha_{11}>\ldots>\alpha_{1r}\) and \(\alpha_{21}<\ldots<\alpha_{2r}\). The main theorem of the paper is that the above equation, satisfying (A), (B), (C) with \(K=(2r+1)^{(r+1)(r+2)}\), has at most \((2r+1)^{r-1}\) non-proportional, primitive solutions. This class of equations includes Thue equations over function fields. The proof applies the \(abc...z\) theorem in function fields. Note that a modified form of Siegel's conjecture over \(\mathbb{Z}\) has been proved by \textit{J. Mueller} and \textit{W. M. Schmidt} [Acta Math. 160, 207-247 (1988; Zbl 0655.10016)] for Thue equations, by a completely different method.
scientific article; zbMATH DE number 1156710
scientific article