𝑆-integral points of ℙⁿ-{2𝕟+1 hyperplanes in general position over number fields and function fields}

Abstract

For the number field case we will give an upper bound on the number of the S S -integral points in P n ( K ) − { 2 n + 1 hyperplanes in general \mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general} position } \text {position}\} . The main tool here is the explicit upper bound of the number of solutions of S S -unit equations (Invent. Math. 102 (1990), 95–107). For the function field case we will give a bound on the height of the S S -integral points in P n ( K ) − { 2 n + 1 hyperplanes in general position } \mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general position}\} . We will also give a bound for the number of “generators" of those S S -integral points. The main tool here is the S S -unit Theorem by Brownawell and Masser (Proc. Cambridge Philos. Soc. 100 (1986), 427–434).