Related papers: Tropical types and associated cellular resolutions
There are many connections between the invariants of the different powers of an ideal. We investigate how to construct minimal resolutions for all powers at once using methods from algebraic and polyhedral topology with a focus on ideals…
A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field. Here, we ask: Where can classical intersection points of planar curves tropicalize to? An answer should have two parts: first, identifying…
This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to…
We investigate location problems whose optimum lies in the tropical convex hull of the input points. Firstly, we study geodesically star-convex sets under the asymmetric tropical distance and introduce the class of tropically quasiconvex…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the…
We consider tropical hemispaces, defined as tropically convex sets whose complements are also tropically convex, and tropical semispaces, defined as maximal tropically convex sets not containing a given point. We introduce the concept of…
We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a higher-dimensional generalization leads to an…
Tropicalization is a procedure for associating a polyhedral complex in Euclidean space to a subvariety of an algebraic torus. We study the question of which graphs arise from tropicalizing algebraic curves. By using Baker's specialization…
We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersurfaces of toric varieties. It follows from these theorems that the integral tropical homology groups of…
Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients.…
To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by…
The prototypical examples of tropical compactifications are compactifications of complements of hyperplane arrangements, which posses a number of remarkable properties not satisfied by more general tropical compactifications of closed…
Starting from certain rational varieties blown-up from (P^1)^N, we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we develop an…
We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian…
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…
Cellular resolutions is a well studied topic on the level of single resolutions and certain specific families of cellular resolutions. One question coming out of the work on families is to understand the structure of cellular resolutions…
This paper is devoted to the bounding and computation of the dimension of deformation spaces of tropical curves and hypersurfaces. This characteristic is interesting in light of the fact that it often coincides with the dimension of…