Related papers: Stiefel tropical linear spaces
Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its…
We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two…
We develop the rudiments of a finite-dimensional representation theory of groups over idempotent semifields by considering linear actions on tropical linear spaces. This can be considered a tropical representation theory, a characteristic…
We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of $A$, which provides the unique right (resp. left) quasi-inverse maximal with respect to the right (resp. left) quasi-identity…
Let $X$ be a closed algebraic subset of $\mathbb{A}^{n}(K)$ where $K$ is an algebraically closed field complete with respect to a nontrivial non-Archimedean valuation. We show that there is a surjective continuous map from the Berkovich…
We consider functions $f:B\to\Rset$ that obey tropical analogs of classical Pl\"ucker relations on minors of a matrix. The most general set $B$ that we deal with in this paper is of the form $\{x\in \Zset^n\colon 0\le x\le a, m\le…
The space T_{d,n} of n tropically collinear points in a fixed tropical projective space TP^{d-1} is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of real d x n matrices of…
We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
In this paper we bring together tropical linear algebra and convex 3-dimensional bodies. We show how certain convex 3-dimensional bodies having 20 vertices and 12 facets can be encoded in a $4\times 4$ integer zero-diagonal matrix $A$. A…
An algorithm to give an explicit description of all the solutions to any tropical linear system $A\odot x=B\odot x$ is presented. The given system is converted into a finite (rather small) number $p$ of pairs $(S,T)$ of classical linear…
This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to…
An unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition. For some particular…
We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. We study its interplay with the general presentation for any finite-dimensional basic algebra. We give an interpretation of evaluating $f_M$ at a weight vector.…
The type A cluster configuration space, commonly known as $\mathcal M_{0,n}$, is the very affine part of the binary geometry associated with the associahedron. The tropicalization of $\mathcal M_{0,n}$ can be realized as the space of…
Flag matroids are combinatorial abstractions of flags of linear subspaces, just as matroids are of linear subspaces. We introduce the flag Dressian as a tropical analogue of the partial flag variety, and prove a correspondence between: (a)…
We describe a canonical compactification of a polyhedral complex in Euclidean space. When the recession cones of the polyhedral complex form a fan, the compactified polyhedral complex is a subspace of a tropical toric variety. In this case,…
Asymptotic properties of matrices are, in general, difficult to analyze with classical mathematical techniques. In very specific cases, there is a well-known connection between the asymptotic behavior of a matrix's leading eigenvector and…
We study the algebraic structure of the semigroup of all $2 \times 2$ tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green's relations and the idempotents and maximal subgroups…
We define the notion of a painted tropical $A$-complex and describe a poset structure on the set of all such complexes. This poset is equivalent to the face lattice of a secondary polytope $\Sigma (\bar{A}_\alpha )$ where $\bar{A}_\alpha$…