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We claim that $M$(atroid) theory may provide a mathematical framework for an underlying description of $M$-theory. Duality is the key symmetry which motivates our proposal. The definition of an oriented matroid in terms of the Farkas…
We show that, for generic bihomogeneous polynomials, the determinant of the matrix of moving planes is irreducible.
The active bijection for oriented matroids (and real hyperplane arrangements, and graphs, as particular cases) is introduced and investigated by the authors in a series of papers. Given any oriented matroid defined on a linearly ordered…
The article considers arrowhead and diagonal-plus-rank-one matrices in F^(nxn) where F in R,C or H. H is a non-commutative field of quaternions. We give unified formulas for fast matrix-vector multiplications, determinants, and inverses for…
A theorem of Mandel allows to determine the covector set of an oriented matroid from its set of topes by using the composition condition. We provide a generalization of that result, stating that the covector set of a conditional oriented…
To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F}\in\mathbb{K}[x]^{n\times n}$ over a field $\mathbb{K}$, we give a deterministic algorithm for finding the determinant of $\mathbf{F}$. The complexity of the algorithm…
Oriented matroids can serve as a tool of modeling of collective decision-making processes in contradictory problems of pattern recognition. We present a generalization of the committee techniques of pattern recognition to oriented matroids.…
We enrich Baker and Bowler's theory of matroids over tracts with notions of vectors and covectors. In the case of oriented matroids, these $F$-vectors and $F$-covectors coincide with the usual signed vectors and signed covectors. In the…
This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension…
This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over GF(2), originally alluded to by Bland and Jensen in their seminal paper on weak…
In this paper, we define the OU matrix of a braid diagram and discuss how the OU matrix reflects the warping degree or the layeredness of the braid diagram, and show that the determinant of the OU matrix of a layered braid diagram is the…
Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential…
We compute hyperdeterminants of hypermatrices whose indices belongs in a meet-semilattice and whose entries depend only of the greatest lower bound of the indices. One shows that an elementary expansion of such a polynomial allows to…
This paper investigates infinite matroids from a categorical perspective. We prove that the category of infinite matroids is a proto-exact category in the sense of Dyckerhoff and Kapranov, thereby generalizing our previous result on the…
In this paper we extend the theory of oriented matroids to Lagrangian orthogonal matroids and their representations, and give a completely natural transformation from a representation of a classical oriented matroid to a representation of…
For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the…
We study matrix factorizations of locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the…
We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of…
L. Lovasz has shown that Sperner's combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. Inspired by this result we prove that classical Ky Fan's theorem admits an…