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Solutions To Systems Of Equations Over Finite Fields, Rachel Petrik
Solutions To Systems Of Equations Over Finite Fields, Rachel Petrik
Theses and Dissertations--Mathematics
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 …