| This dissertation is on sums-of-squares formulas, focusing on whether existence of a formula depends on the base field and on methods for finding sums-of-squares formulas.;We introduce a new approach to the study of sums-of-squares formulas, showing that a formula can be regarded as a solution to a system of polynomial equations and thus we can introduce the variety of sums-of-squares formula. This new perspective allows us to utilize tools of algebraic geometry in the study of sums-of-squares formulas, as well as raising new questions about the properties of this variety.;We use number theory and computational algebraic geometry to consider the question of whether existence of sums-of-squares formulas depends on the base field. We are able to show independence in some cases, however, the general case remains an open question. Furthermore, we show that existence of a formula over an algebraically closed field is computable.;We introduce an algebraic group action on the variety of sums-of-squares formulas, which gives us another way to study the structure of this variety and raises new questions about the structure of the action.;Finally, we provide algorithms for finding sums-of-squares formulas over the integers and finite fields. These algorithms use previous results about formulas over the integers, as well as the group action on the scheme of sums-of-squares formulas. |
