Solutions to Systems of Equations over Finite Fields

This dissertation investigates the existence of solutions to equations over finite fields with an emphasis on diagonal equations. In particular: Given a system of equations, how many solutions are there? In the case of a system of diagonal forms, when does a nontrivial solution exist? Many results are known that address (1) and (2), such as the classical Chevalley--Warning theorems. With respect to (1), we have improved a recent result of D.R. Heath--Brown, which provides a lower bound on the total number of solutions to a system of polynomials equations. Furthermore, we have demonstrated that several of our lower bounds are sharp under the stated hypotheses. With respect to (2), we have several improvements that extend known results. First, we have improved a result of James Gray by extending his theorem to a larger class of equations. Second, for particular degrees, number of forms, and finite fields, we have determined the minimal number of variables needed to guarantee the existence of a nontrivial solution. Third, there are many results, which address (2) for particular types of systems known as A-systems. We give a criterion that characterizes when a system of equations is an A-system. Finally, we have provided exposition that adds significantly more detail to two important papers by Tietäväinen