Dissertation
University of Utah
Patrick Dylan Zwick
This is where you can find my Ph.D. dissertation, along with the Maple worksheet files used to perform the computations needed for that dissertation.
Variations on a Theme of Symmetric Tropical Matrices
Abstract - 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 x 4 minors of an n x 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 x r minors of an n x 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 x 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.
The Files
The Dissertation
Variations on a Theme of Symmetric Tropical Matrices
The Maple Code
Below are the Maple worksheets and Maple library archive files used to verify the computational claims made in Chapter 5 of the dissertation. These worksheets are described in the dissertation's second appendix.
Please note that I make no claims about the efficiency of these Maple procedures. It's quite possible there are more efficient ways of implementing them. The only claim I make (and even here, I wouldn't bet my life) is that the procedures work correctly. Also, the only reason I used Maple is that I had some experience with it, and it was available. It's quite possible these procedures would be more useful if they had been written in a different language. The version of Maple used was Maple 17.00.
Tropical Rank Calculations
The Maple archive file with the TropLinAlg package - TropLinAlg.mla
The worksheet used to verify the computational claims made in Chapter 5 concerning tropical and symmetric tropical rank - TropicalRankCalculations.mw
The worksheet used to create the package TropLinAlg and export it as a .mla file - TropicalRankModule.mw
Local Dimension Lower Bounds Calculations
The Maple archive file with the TropLinAlgLocalDimensionLowerBounds package - TropLinAlgLocalDimensionLowerBounds.mla
The worksheet used to verify the computational claims made in Chapter 5 concerning lower bounds on local dimensions of determinantal tropical prevarieties - LocalDimensionLowerBoundsCalculations.mw
The worksheet used to create the package TropLinAlgLocalDimensionLowerBounds and export it as a .mla file - TropLinAlgLocalDimensionLowerBoundsModule.mw
The Defense
Here are the Beamer slides for my dissertation defense - Dissertation Defense
If you have any issues with any of these files, or any comments in general, please email me at dylanzwick@gmail.com
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