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65
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 ..."
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Cited by 125 (10 self)
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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.
Duality and separation theorems in idempotent semimodules
 Linear Algebra and its Applications 379 (2004), 395–422. Also arXiv:math.FA/0212294
"... Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to sep ..."
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Cited by 67 (20 self)
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Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and halfspaces over the maxplus semiring. 1.
The Minkowski Theorem for Maxplus Convex Sets
, 2006
"... We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones a ..."
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Cited by 36 (14 self)
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We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones and closed unbounded maxplus convex sets. In particular, we show that a closed maxplus convex set can be decomposed as a maxplus sum of its recession cone and of the maxplus convex hull of its extreme points.
Tropical polyhedra are equivalent to mean payoff games
 Int. J. of Algebra and Computation, Eprint
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Inferring Min and Max Invariants Using Maxplus Polyhedra
"... Abstract. We introduce a new numerical abstract domain able to infer min and max invariants over the program variables, based on maxplus polyhedra. Our abstraction is more precise than octagons, and allows to express nonconvex properties without any disjunctive representations. We have defined sou ..."
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Cited by 17 (7 self)
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Abstract. We introduce a new numerical abstract domain able to infer min and max invariants over the program variables, based on maxplus polyhedra. Our abstraction is more precise than octagons, and allows to express nonconvex properties without any disjunctive representations. We have defined sound abstract operators, evaluated their complexity, and implemented them in a static analyzer. It is able to automatically compute precise properties on numerical and memory manipulating programs such as algorithms on strings and arrays. 1
The number of extreme points of tropical polyhedra
, 2009
"... The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in ..."
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Cited by 16 (8 self)
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The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in the tropical setting, up to a trivial modification. Then, we study the natural candidates to be the maximizing polyhedra, which are the polars of a family of cyclic polytopes equipped with a sign pattern. We construct bijections between the extreme points of these polars and lattice paths depending on the sign pattern, from which we deduce explicit bounds for the number of extreme points, showing in particular that the upper bound is asymptotically tight as the dimension tends to infinity, keeping the number of constraints fixed. When transposed to the classical case, the previous constructions yield some lattice path generalizations of Gale’s evenness criterion.
Factorization of polynomials in one variable over the tropical semiring
, 2005
"... We show factorization of polynomials in one variable over the tropical semiring is in general NPcomplete, either if all coefficients are finite, or if all are either 0 or infinity (Boolean case). We give algorithms for the factorization problem which are not polynomial time in the degree, but are p ..."
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Cited by 14 (1 self)
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We show factorization of polynomials in one variable over the tropical semiring is in general NPcomplete, either if all coefficients are finite, or if all are either 0 or infinity (Boolean case). We give algorithms for the factorization problem which are not polynomial time in the degree, but are polynomial time for polynomials of fixed degree. For twovariable polynomials we derive an irreducibility criterion which is almost always satisfied, even for fixed degree, and is polynomial time in the degree. We prove there are unique least common multiples of tropical polynomials, but not unique greatest common divisors. We show that if two polynomials in one variable have a common tropical factor, then their eliminant matrix is singular in the tropical sense. We prove the problem of determining tropical rank is NPhard.
Maxplus convex geometry
 of Lecture Notes in Comput. Sci
, 2006
"... Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theore ..."
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Cited by 14 (9 self)
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Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theorem, and the characterization of the analogues of “simplicial ” cones in terms of distributive lattices. 1