Results 1  10
of
168
Algorithms in Discrete Convex Analysis
 Math. Programming
, 2000
"... this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects. ..."
Abstract

Cited by 158 (36 self)
 Add to MetaCart
(Show Context)
this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects.
Tropical geometry and its applications
 International Congress of Mathematicians vol. II, 827–852, Eur. Math. Soc
, 2006
"... Abstract. From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewiselinear objects that take over the rôle of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the pr ..."
Abstract

Cited by 141 (6 self)
 Add to MetaCart
(Show Context)
Abstract. From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewiselinear objects that take over the rôle of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the problems in classical (real and complex) geometry.
First Steps in Tropical Geometry
 CONTEMPORARY MATHEMATICS
"... Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete descr ..."
Abstract

Cited by 123 (10 self)
 Add to MetaCart
Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus’ Theorem.
The Bergman complex of a matroid and phylogenetic trees
 THE JOURNAL OF COMBINATORIAL THEORY, SERIES B
, 2005
"... ..."
Tropical discriminants
, 2005
"... Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel’fand, Kapranov and Zelevinsky. The tropical Adiscriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown t ..."
Abstract

Cited by 62 (6 self)
 Add to MetaCart
(Show Context)
Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel’fand, Kapranov and Zelevinsky. The tropical Adiscriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalization of the kernel of A. This leads to an explicit positive formula for the extreme monomials of any Adiscriminant, and to a combinatorial rule for deciding when two regular triangulations of A correspond to the same monomial of the Adiscriminant.
MATROID POLYTOPES, NESTED SETS AND BERGMAN FANS
, 2004
"... The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial com ..."
Abstract

Cited by 55 (6 self)
 Add to MetaCart
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of ArdilaKlivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De ConciniProcesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences.
On the rank of a tropical matrix
, 2005
"... This is a foundational paper in tropical linear algebra, which is linear algebra over the minplus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally ..."
Abstract

Cited by 55 (5 self)
 Add to MetaCart
(Show Context)
This is a foundational paper in tropical linear algebra, which is linear algebra over the minplus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally in polyhedral and algebraic geometry, and we show that they differ in general. Realizability of matroids plays a crucial role here. Connections to optimization are also discussed.
Analytification is the limit of all tropicalizations
"... Abstract. We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings. 1. ..."
Abstract

Cited by 51 (7 self)
 Add to MetaCart
(Show Context)
Abstract. We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings. 1.
Tropical linear spaces
, 2004
"... We define tropical analogues of the notions of linear space and Plücker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main ..."
Abstract

Cited by 49 (2 self)
 Add to MetaCart
(Show Context)
We define tropical analogues of the notions of linear space and Plücker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main result that all constructible tropical linear spaces have the same fvector and are “seriesparallel”. We conjecture that this fvector is maximal for all tropical linear spaces with equality precisely for the seriesparallel tropical linear spaces. We present many partial results towards this conjecture. In addition we describe the relation of tropical linear spaces to linear spaces defined over power series fields and give many examples and counterexamples illustrating aspects of this relationship. We describe a family of particularly nice seriesparallel linear spaces, which we term tree spaces, that realize the conjectured maximal fvector and are constructed in a manner similar to the cyclic polytopes.