A Tropical Toolkit
TLDR
In this paper, the authors give an introduction to tropical geometry and prove some results in tropical intersection theory and give a foundational account of intersection theory with proofs of new theorems relating it to classical intersection theory.About:
This article is published in Expositiones Mathematicae.The article was published on 2009-01-01 and is currently open access. It has received 76 citations till now. The article focuses on the topics: Toric variety & Intersection theory.read more
Citations
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Journal ArticleDOI
First Steps in Tropical Intersection Theory
Lars Allermann,Johannes Rau +1 more
TL;DR: In this paper, a tropical intersection theory of cycles, Cartier divisors, and intersection products between cycles and cycles is established, without passing to rational equivalence, and the push-forward and pull-back of this theory is discussed.
Journal ArticleDOI
On the tropical Torelli map
TL;DR: In this paper, the moduli spaces of tropical curves and tropical principally polarized abelian varieties are constructed in the category of stacky fans, and the tropical Torelli map between these two modulus spaces is defined.
Journal ArticleDOI
Log-concavity of characteristic polynomials and the Bergman fan of matroids
TL;DR: In this article, the authors proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory.
Posted Content
A guide to tropicalizations
TL;DR: In this article, a theory of toric schemes over valuation rings of rank 1 was developed for algebraic and convex problems, where the toric co-ordinates are well suited to the convex problem, and it is sometimes possible to use a solution of a convex solution to solve the original algebraic problem.
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Lifting Tropical Intersections
Brian Osserman,Sam Payne +1 more
TL;DR: In this paper, it was shown that points in the intersection of the tropicaliza-tions of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension.
References
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Book
Geometric Invariant Theory
TL;DR: Geometric invariant theory for moduli spaces has been studied extensively in the mathematical community as mentioned in this paper, with a large number of applications to the moduli space construction problem, see, for instance, the work of Mumford and Fogarty.
Book
Discriminants, Resultants, and Multidimensional Determinants
TL;DR: The Cayley method of studying discriminants was used by Cayley as discussed by the authors to study the Cayley Method of Discriminants and Resultants for Polynomials in One Variable and for forms in Several Variables.
Book
Gröbner bases and convex polytopes
TL;DR: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry as mentioned in this paper.
Book
Rational Curves on Algebraic Varieties
TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.
MonographDOI
Solving systems of polynomial equations
TL;DR: Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials as mentioned in this paper are the primary decomposition of polynomial systems in economics and statistics.