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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 142 (7 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 24 (2 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.
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 21 (6 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.
Supertropical algebra
, 2007
"... Abstract. We develop the algebraic polynomial theory for “supertropical algebra, ” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more gener ..."
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Cited by 16 (9 self)
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Abstract. We develop the algebraic polynomial theory for “supertropical algebra, ” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more general situation over an arbitrary semiring, and develop a theory which contains the analogs of the basic theorems of classical commutative algebra (such as the Euclidean algorithm and the Hilbert Nullstellensatz), as well as some results without analogs in the classical theory, such as generation of prime ideals of the polynomial semiring by binomials. Examples are also given to show how this theory differs from classical commutative algebra.
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 15 (9 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 15 (8 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.
What shape is your conjugate? A survey of computational convex analysis and its applications
 SIAM Rev
, 2010
"... 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 ..."
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Cited by 9 (3 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 useful for applications in image processing (computing the distance transform, the generalized distance transform, and mathematical morphology operators), partial differential equations (solving HamiltonJacobi equations, and using differential equations numerical schemes to compute the convex envelope), maxplus algebra (computing the equivalent of the Fast Fourier Transform), multifractal analysis, etc. The fields of applications include, among others, computer vision, robot navigation, thermodynamics, electrical networks, medical imaging, and network communication.
TROPICAL ALGEBRAIC SETS, IDEALS AND AN ALGEBRAIC NULLSTELLENSATZ
, 2008
"... This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring – a structure that ..."
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Cited by 8 (2 self)
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This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring – a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra. The use of the new notion of tropical algebraic comsets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz.