The truncated second main theorem of function fields
This paper proves the function field analogue of Cartan's second main theorem with truncated counting functions in arbitrary characteristic, and the Nochka-Chen theorem (Cartan's conjecture) with truncated counting functions in characteristic zero. In both of these cases, the hyperplanes are required to be defined over the field of constants of the function field. The paper also generalizes the higher-dimensional \(abc\) theorem of Voloch and Brownawell-Masser to characteristic \(p>0\). As a corollary of the Nochka-Chen theorem, the paper extends the result of \textit{M. Ru} and \textit{P.-M. Wong} [Invent. Math. 106, 195-216 (1991; Zbl 0758.14007)] on finiteness of integral points on \(\mathbb{P}^n\) minus \(2n+1\) hyperplanes in general position, to function fields of characteristic zero. Again, the hyperplanes must be defined over the field of constants of the function field. Numerous corollaries are also given, and all results include effective, explicit constants.
scientific article; zbMATH DE number 883842