Combinatorial aspects of tropical geometry.

摘要

Tropical convex geometry and tropical algebraic geometry arise from linear and polynomial algebra over the tropical semiring ( R , min, +). They also appear as images of convex and algebraic sets over fields with valuations into the real numbers. In this thesis, the combinatorial aspects of tropical geometry are studied through three topics: tropical polytopes, tropical linear spaces, and tropical elimination and implicitization.;Tropical polytopes are related to cellular resolutions of monomial ideals in two different ways. First, their natural polyhedral complex structure supports minimal linear free resolutions of monomial initial ideals of the ideals of 2 x 2-minors of matrices of unknowns. Secondly, for an arbitrary monomial ideal, the tropical convex hull of the exponents of its minimal generators is the common shadow of a natural family of cellular free resolutions, generalizing the hull resolution.;Tropical linear spaces are some special types of intersections of tropical hyperplanes. We show that they are also images of certain tropical linear maps. We characterize the sets of tropical hyperplanes and the parametrizations that define a tropical linear space. This generalizes the problem of finding minimal tropical bases of matroids. We show that graphic and cographic matroids have unique minimal tropical bases.;Elimination and implicitization are computational problems of finding the defining ideals of a projection of an algebraic variety and a variety given by a parametrization, respectively. We give a combinatorial construction of the tropicalization of these ideals when the given input polynomials have generic coefficients. When the solution is a principal ideal, we can recover the Newton polytope of the generating polynomial from the tropical variety, and in general we can compute its Chow polytope. We also describe an implementation of this method.