In this talk, we'll study conditions which guarantee that a diagonal homogeneous polynomial has a zero over a field of p-adic numbers. I'll begin by giving an overview of what the p-adic numbers are and how they are constructed, mentioning in particular some properties of p-adic numbers which differ from those of the real numbers. Then I'll talk about homogeneous polynomials over the p-adics and present some theorems. I may outline some proofs, but in general this presentation will focus on ideas rather than technical details.
This talk will be accessible to Juniors and Seniors.