Related papers: Tropical geometry of PERT
This document is a slightly expanded version of a series of talks given by J. Giansiracusa at the workshop `Geometry over semirings' at Universitat Aut\`{o}noma de Barcelona in July 2025. In the first lecture we introduce tropical…
We introduce the first graph kernels for metric graphs via tropical algebraic geometry. In contrast to conventional graph kernels based on graph combinatorics such as nodes, edges, and subgraphs, our metric graph kernels are purely based on…
This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $\nu$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over $\nu$-semirings, whose distinguished ghost…
We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the…
We investigate the Voronoi diagrams with respect to an asymmetric tropical distance function, also for infinite point sets. These turn out to be much better behaved than the tropical Voronoi diagrams arising from the standard tropical…
Tropical algebraic geometry offers new tools for elimination theory and implicitization. We determine the tropicalization of the image of a subvariety of an algebraic torus under any homomorphism from that torus to another torus.
A complex network approach is proposed to study the shear behavior of a rough rock joint. Similarities between aperture profiles are established and a general network in two directions (in parallel and perpendicular to the shear direction)…
This is a guide on how to create 3d printable models of tropical surfaces, curves, and combinations thereof. It uses Polymake to construct bounded models of the tropical objects, and OpenSCAD to thicken and export them to any common 3D…
We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using…
Using tropical geometry one can translate problems in enumerative geometry to combinatorial problems. Thus tropical geometry is a powerful tool in enumerative geometry over the complex and real numbers. Results from $\mathbb{A}^1$-homotopy…
A brief introduction to tropical and idempotent mathematics (with an emphasys on idempotent functional analysis) is presented. Applications to classical mechanics and geometry are especially examined.
The topological complexity ${\sf TC}(X)$ is a homotopy invariant of a topological space $X$, motivated by robotics, and providing a measure of the navigational complexity of $X$. The topological complexity of a connected sum of real…
We study the combinatorial properties of 2-dimensional tropical complexes. In particular, we prove tropical analogues of the Hodge index theorem and Noether's formula. In addition, we introduce algebraic equivalence for divisors on tropical…
In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of…
As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R,min,+). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels in a…
We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We…
We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these…
Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the…
We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.