Related papers: Linearizing Algebraic Matroids
Inspired by Kontsevich's graphic orbifold Euler characteristic we define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank $r$. Our main result provides a simple formula for the virtual Euler…
We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular we study the representability of its dual, providing an extension of the Gale duality…
We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…
A $q$-matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These $q$-matroids are motivated by coding theory as the representable…
Attribute reduction is a basic issue in knowledge representation and data mining. Rough sets provide a theoretical foundation for the issue. Matroids generalized from matrices have been widely used in many fields, particularly greedy…
We extend the notion of matroid representations by matrices over fields and consider new representations of matroids by matrices over finite semirings, more precisely over the boolean and the superboolean semirings. This idea of…
We call a linear operator on a vector space over a field Jordanable if it has a Jordan canonical form. An almost Abelian Lie algebra has only one non-vanishing Lie bracket, which is given by a linear operator. If the latter is Jordanable…
We introduce a new representation concept for lattices by boolean matrices, and utilize it to prove that any matroid is boolean representable. We show that such a representation can be easily extracted from a representation of the…
An arithmetic matroid is weakly multiplicative if the multiplicity of at least one of its bases is equal to the product of the multiplicities of its elements. We show that if such an arithmetic matroid can be represented by an integer…
A number of models of linear logic are based on or closely related to linear algebra, in the sense that morphisms are "matrices" over appropriate coefficient sets. Examples include models based on coherence spaces, finiteness spaces and…
We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory…
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…
We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. For instance, $A$ be a linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can see that…
Though it goes without saying that linear algebra is fundamental to mathematical biology, polynomial algebra is less visible. In this article, we will give a brief tour of four diverse biological problems where multivariate polynomials play…
We assess the situation of our elementary Linear Algebra classes in the US holistically and through personal history recollections. Possible remedies for our elementary Linear Algebra's teaching problems are discussed and a change from…
In this short note, we give a new sufficient condition for a linear map from a product of copies of a field to endomorphisms of a finite dimensional vector space over the same field to be an algebra homomorphism. We expect that this result…
This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if…
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…
Linear information and rank inequalities as, for instance, Ingleton inequality, are useful tools in information theory and matroid theory. Even though many such inequalities have been found, it seems that most of them remain undiscovered.…
A class of matroids is introduced which is very large as it strictly contains all paving matroids as special cases. As their key feature these split matroids can be studied via techniques from polyhedral geometry. It turns out that the…