Resultants and height bounds for zeros of homogeneous polynomial systems

In 1955, Cassels proved a now celebrated theorem giving a search bound algorithm for determining whether a quadratic form has a nontrivial zero over the rationals. Since then, his work has been greatly generalized, but most of these newer techniques do not follow his original method of proof. In this thesis, we revisit his 1955 proof, modernize his tools and language, and use this machinery to prove more general theorems regarding height bounds for the common zeros of a system of polynomials in terms of the heights of those polynomials. We then use these theorems to give a short proof of a more general (albeit, known) version of Cassels' Theorem and give some weaker results concerning the rational points of a cubic or a pair of quadratics.