Generalizing Waring Problem & Weil Conjecture: On Dimension & Existence of Special Points on Schemes of Higher Degree

Abstract

In this thesis, we analyze relationship between number of variables & the existence of special values over higher dimensional surfaces defined over various fields. This thesis is roughly divided into three parts. In the first part, we use classical techniques in additive number theory to upper bound the number of variables needed to guarantee the existence of points over Fp, Zp, Qp and Q. Next, we switch gear and looking at modern algebra & algebraic geometry techniques. We offering a generalization of Weil’s conjecture for special set of In particular, we look at results involving Brauer groups, and offered a discussion of the connection between Brauer group, and birationality among curves. Finally, we offer a brief discussion on Sarnak’s involution methods, in providing a generic bound for symmetric surfaces that is not necessarily homogeneous.