Related papers: Tropical computations for toric intersection theor…
We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing…
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…
We define arroids as an abstract axiom set encoding the intersection properties of arrangements of curves. The tropicalization of the complement of arrangement of curves meeting pairwise transversely is shown to be determined by the…
Tropical R is the birational map that intertwines products of geometric crystals and satisfies the Yang-Baxter equation. We show that the D^{(1)}_n tropical R introduced by the authors and its reduction to A^{(2)}_{2n-1} and C^{(1)}_n are…
Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric…
We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack ; to obtain an explicit…
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how…
The paper studies intrinsic geometry in the tropical plane. Tropical structure in the real affine $n$-space is determined by the integer tangent vectors. Tropical isomorphisms are affine transformations preserving the integer lattice of the…
We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev…
We employ tropical algebras as platforms for several cryptographic schemes that would be vulnerable to linear algebra attacks were they based on "usual" algebras as platforms.
We define and study the cyclic Bergman fan of a matroid M, which is a simplicial polyhedral fan supported on the tropical linear space T(M) of M and is amenable to computational purposes. It slightly refines the nested set structure on…
This paper is the third installment in a series of papers devoted to the computation of enumerative invariants of abelian surfaces through the tropical approach. We develop a pearl diagram algorithm similar to the floor diagram algorithm…
We introduce the tropicalization of closed subschemes of a torus defined over a higher dimensional local field. We study the basic invariants of such tropicalizations. This is a generalization of the results of Einslieder, Kapranov, Lind,…
We introduce ParaToric, a C++ package for simulating the toric code in a parallel field (i.e., $X$- and $Z$-fields) at finite temperature. We implement and extend the continuous-time quantum Monte Carlo algorithm of Wu, Deng, and Prokof'ev…
This paper investigates the geometric properties of a special case of the two-sided system given by $2 \times 2$ tropical commuting constraints. Given a finite matrix $A \in \mathbb{R}^{2\times 2}$, the paper studies the extremals of the…
We construct an analog of the Hodge theory on complex manifolds on tropical curves. We use the analytical approach to the problem, it is based on language of tropical differential forms and methods of $L^2-$cohomologies.
We use extensions of tropical algebras as platforms for very efficient public key exchange protocols.
We survey some results on toric topology.
We present TropNNC, a framework for compressing neural networks with linear and convolutional layers and ReLU activations using tropical geometry. By representing a network's output as a tropical rational function, TropNNC enables…
We develop a tropical intersection formalism of forms and currents that extends classical tropical intersection theory in two ways. First, it allows to work with arbitrary polytopes, also non-rational ones. Second, it allows for smooth…