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In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Gröbner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plücker relations. It is shown to parametrize all tropical(More)
In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Grk,n)≥0. This is a cell complex whose cells G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known(More)
This dissertation is dedicated to the memory of my mother, Kathleen Marie Harter. ii ACKNOWLEDGEMENTS I have come this far largely because I have had a great deal of support, encouragement , and companionship from amazing teachers, friends, and colleagues, and I would like to take this opportunity to thank some of those people. Graduate school at Michigan(More)
Tropical Geometry by David E Speyer Doctor of Philosophy in Mathematics University of California, Berkeley Professor Bernd Sturmfels, Chair Let K be an algebraically closed field complete with respect to a nonarchimedean valuation v : K∗ → R. The reader should think ofK as the field of Puiseux series, ⋃∞ n=1 C((t )) and v as the map that assigns to a power(More)
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