We apply inequalities from the theory of linear forms in logarithms to deduce effective results on -integral points on certain higher-dimensional varieties when the cardinality of is sufficiently small. These results may be viewed as a higher-dimensional version of an effective result of Bilu on integral points on curves. In particular, we prove a completely explicit result for integral points on certain affine subsets of the projective plane. As an application, we generalize an effective result of Vojta on the three-variable unit equation by giving an effective solution of the polynomial unit equation , where , , and are -units, , and is a polynomial satisfying certain conditions (which are generically satisfied). Finally, we compare our results to a higher-dimensional version of Runge’s method, which has some characteristics in common with the results here.
"Linear forms in logarithms and integral points on higher-dimensional varieties." Algebra Number Theory 8 (3) 647 - 687, 2014. https://doi.org/10.2140/ant.2014.8.647