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.

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 half-spaces over the max-plus semiring. 1.

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Abstract. We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if K is a conditionally complete idempotent semifield, with completion ¯ K, a convex function K n → ¯ K which is lower semi-continuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation theorem for closed convex subsets of K n, which extends earlier results of Zimmermann, Samborski, and Shpiz.

Abstract. We establish the following max-plus analogue of Minkowski’s theorem. Any point of a compact max-plus convex subset of (R ∪ {−∞}) n can be written as the max-plus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed max-plus convex cones and closed unbounded max-plus convex sets. In particular, we show that a closed max-plus convex set can be decomposed as a max-plus sum of its recession cone and of the max-plus convex hull of its extreme points. 1.

We show factorization of polynomials in one variable over the tropical semiring is in general NP-complete, 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 two-variable 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 NP-hard. 1

Abstract. We introduce a new numerical abstract domain able to infer min and max invariants over the program variables, based on max-plus polyhedra. Our abstraction is more precise than octagons, and allows to express non-convex 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

Abstract—This paper formally models and studies latencyinsensitive systems (LISs) through max-plus algebra. We introduce state traces to model behaviors of LISs and obtain a formally proved performance upper bound achievable by latencyinsensitive design. An implementation of the latency-insensitive protocol that can provide robust communication through backpressure is also proposed. The intrinsic performance of the proposed implementation is acquired based on state traces. It is also proved that the proposed implementation can always reach the best performance achievable by latency-insensitive design. Index Terms—Back-pressure, latency-insensitive system, maxplus algebra, performance analysis, state trace.

Abstract. Max-plus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including max-plus versions of the separation theorem, existence of linear and non-linear projectors, max-plus analogues of the Minkowski-Weyl theorem, and the characterization of the analogues of “simplicial ” cones in terms of distributive lattices. 1

Abstract. 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. 1.