Related papers: A Grassmann algebra for matroids
The isotropic Grassmannian parametrizes isotropic subspaces of a vector space equipped with a quadratic form. In this paper, we show that any maximal isotropic Grassmannian in its Pl\"ucker embedding can be defined by pulling back the…
Any algebra herein is intended over a field of characteristic 0. Let $E$ denote the infinite dimensional Grassman algebra. Given a power associative finite dimensional {$\mathbb{Z}_2$-graded-central-simple} $A$ and a supertrace algebra $B$,…
In this paper we give a necessary and sufficient criterion for representability of a matroid over an algebraic closed field. This leads to an algorithm, based on an extension of Groebner Bases, in order to decide if a given matroid is…
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…
The Weil reciprocity law asserts that given two meromorphic functions $f, g$ on a compact complex curve, the product of the values of $f$ over the roots and poles of $g$ is equal to the product of the values of $g$ over the roots and poles…
Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of $\mathrm{PGL}$ parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector…
We develop a novel framework to construct and analyze finite valued, multidimensional mechanisms using tropical convex geometry. We geometrically characterize incentive compatibility using cells in the tropical convex hull of the type set.…
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…
In this paper we discuss the dimensions of the (higher) secant varieties to the Grassmann varieties, embedded via the Plucker embeddings. We use Terracini's Lemma and the duality in the exterior algebra of a finite dimensional vector space…
In this paper, we study tropical Weierstrass points. These are the analogues for tropical curves of ramification points of line bundles on algebraic curves. For a divisor on a tropical curve, we associate intrinsic weights to the connected…
We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical…
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…
The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup…
A Laman graph $G$ is a minimally rigid graph in dimension two, and its realization number is its number of distinct embeddings with fixed generic edge lengths. While conjectured to grow exponentially in the number of vertices of $G$, the…
A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree tensor networks is proposed and analyzed. In a recursion from the leaves to the root, the integrator updates bases and then evolves…
We undertake a combinatorial study of the piecewise linear map g : R^{2m+2n} --> R^{mn} which assigns to the four vectors a, A in R^m and b, B in R^n the m by n matrix given by g_{ij} = min (a_i + b_j, A_i+B_j). This map arises naturally in…
We study linear degenerations of flag varieties from the point of view of tropical geometry. We define the linear degenerate flag Dressian and prove a correspondence between: $(a)$ points in the linear degenerate flag Dressian, $(b)$ linear…
In many instances one has to deal with parametric models. Such models in vector spaces are connected to a linear map. The reproducing kernel Hilbert space and affine- / linear- representations in terms of tensor products are directly…
We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most…