| Description |
Tropical geometry connects the fields of algebraic and polyhedral geometry. This connection has been used to discover much simpler proofs of fundamental theorems in algebraic geometry, including the Brill-Noether theorem. Tropical geometry has also found applications outside of pure mathematics, in areas as diverse as phylogenetic models and auction theory. This dissertation seeks to answer the question of when the minors of a symmetric matrix form a tropical basis. The first chapter introduces the relevant ideas and concepts from tropical geometry and tropical linear algebra. The second chapter introduces different notions of rank for symmetric tropical matrices. The third chapter is devoted to proving all the cases, outside symmetric tropical rank three, where the minors of a symmetric matrix form a tropical basis. The fourth chapter deals with symmetric tropical rank three. We prove that the 4 × 4 minors of an n×n symmetric matrix form a tropical basis if n ≤ 5, but not if n ≥ 13. The question for 5 < n < 13 remains open. The fifth chapter is devoted to when the minors of a symmetric matrix do not form a tropical basis. We prove the r × r minors of an n × n symmetric matrix do not form a tropical basis when 4 < r < n. We also prove that, when the minors of a matrix (general or symmetric) define a tropical variety and tropical prevariety that are different, then, with one exception, the two sets differ in dimension. The exception is the 4 × 4 minors of a symmetric matrix, where the question is still unresolved. The sixth chapter explores tropical conics. A correspondence between a property of the symmetric matrix of a quadric and the dual complex of that quadric is demonstrated for conics, and proposed for all quadrics. The seventh chapter reviews the results and proposes possible questions for further study. The first appendix is devoted to correcting a proof in a paper cited by this dissertation. The second appendix is a transcript of the Maple worksheets used to perform the computer calculations from the fifth chapter. |
