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Tropical geometry and its applications
 the Proceedings of the Madrid ICM
"... Abstract. These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM. 1. ..."
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Cited by 61 (3 self)
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Abstract. These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM. 1.
Moduli spaces of rational tropical curves
 Proceedings of Gökova GeometryTopology Conference 2006, 39–51, Gökova Geometry/Topology Conference (GGT), Gkova
, 2007
"... Abstract. This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n − 3. We define the DeligneMumford compactification of this space and tropical ψclass divisors. Thi ..."
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Cited by 11 (1 self)
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Abstract. This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n − 3. We define the DeligneMumford compactification of this space and tropical ψclass divisors. This paper gives a detailed description of the moduli space of tropical rational curves mentioned in [4]. The survey [4] was prepared under rather sharp time and volume constraints. As a result the coordinate presentation of this moduli space from [4] contains a mistake (it was oversimplified). In this paper we’ll correct the mistake and give a detailed description on M0,5 as our main example.
Tropical arithmetic & algebra of tropical matrices
, 2005
"... The purpose of this paper is to study the tropical algebra – the algebra over the tropical semiring. We start by introducing a new approach to arithmetic over the maxplus semiring which generalizes the former concept in use. Regarding this new arithmetic, matters of tropical matrices are discusse ..."
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Cited by 9 (6 self)
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The purpose of this paper is to study the tropical algebra – the algebra over the tropical semiring. We start by introducing a new approach to arithmetic over the maxplus semiring which generalizes the former concept in use. Regarding this new arithmetic, matters of tropical matrices are discussed and the properties of these matrices are studied. These are the preceding phases toward the characterization of the tropical inverse matrix which is eventually attained. Further development yields the notion of tropical normalization and the principle of basis change in the tropical sense.
Cyclic projectors and separation theorems in idempotent convex geometry
 Journal of Mathematical Sciences
, 2008
"... Semimodules over idempotent semirings like the maxplus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the nfold cartesian product of the maxplus semiring: it is known that one can separate a vector from a closed s ..."
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Cited by 6 (5 self)
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Semimodules over idempotent semirings like the maxplus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the nfold cartesian product of the maxplus semiring: it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed subsemimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearestpoint projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert’s projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly’s theorem.
Tropical varieties, ideals and an algebraic nullstellensatz. Preprint at arXiv:math.AC/0511059
, 2005
"... The objective of this paper is to introduce the fundamental algebrogeometric constructions over the extended tropical semiring. The study of tropical varieties, covarieties and ideals over this extension eventually yields the theorem of the weak tropical Nullstellensatz and gives an algebraic inte ..."
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Cited by 5 (2 self)
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The objective of this paper is to introduce the fundamental algebrogeometric constructions over the extended tropical semiring. The study of tropical varieties, covarieties and ideals over this extension eventually yields the theorem of the weak tropical Nullstellensatz and gives an algebraic interpretation of the tropical Nullstellensatz.
WHAT SHAPE IS YOUR CONJUGATE? A SURVEY OF COMPUTATIONAL CONVEX ANALYSIS AND ITS APPLICATIONS
"... Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental t ..."
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Cited by 5 (1 self)
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Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolicnumeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms, and to further communications between several fields benefiting from the same techniques. We survey applications of the algorithms which have been applied to problems arising from image processing (distance transform, generalized distance transform, mathematical morphology), partial differential equations (solving HamiltonJacobi equations, and using differential equations numerical schemes to compute the convex envelope), maxplus algebra, multifractal analysis, and several others. They span a wide range of applications in computer vision, robot navigation, phase transition in thermodynamics, electrical networks,
The tropical rank of a tropical matrix
 Communications in Algebra
"... In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependency allows for a natural definition of matrix rank in a sense that coincides with the notion of tropical regularity. ..."
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Cited by 3 (0 self)
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In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependency allows for a natural definition of matrix rank in a sense that coincides with the notion of tropical regularity.
Tropical analysis on plurisubharmonic singularities
 In this volume (part 2
"... Tropical structures appear naturally in investigation of singularities of plurisubharmonic functions. We show that standard characteristics of the singularities can be viewed as tropicalizations of certain notions from commutative algebra. In turn, such a consideration gives a tool for studying the ..."
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Cited by 2 (0 self)
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Tropical structures appear naturally in investigation of singularities of plurisubharmonic functions. We show that standard characteristics of the singularities can be viewed as tropicalizations of certain notions from commutative algebra. In turn, such a consideration gives a tool for studying the singularities. In addition, we show how the notion of Newton polyhedron and its generalizations come into the picture as a result of the tropicalization. 1
Tropical Algebraic Geometry in Maple  a preprocessing algorithm for finding common factors to multivariate polynomials with approximate coefficients
, 2009
"... ..."
On commuting matrices in max algebra and in classical nonnegative algebra
, 2010
"... This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, pa ..."
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This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described, and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode.