Related papers: Matroidal Structure of Skew Polynomial Rings with …
To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration. In this note we observe that…
We investigate the universality of microscopic eigenvalue correlations for Random Matrix Theories with the global symmetries of the QCD partition function. In this article we analyze the case of real valued chiral Random Matrix Theories…
The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. We introduce a method to study these strata by way of the infinite-area translation surface associated to the…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
Two are the objectives of the present paper. First we study properties of a differentially simple commutative ring R with respect to a set D of derivations of R. Among the others we investigate the relation between the D-simplicity of R and…
We study the interplay between the lattice of F_{q^m}-subspaces and the lattice of F_{q^m}-subspaces of an F_{q^m}-vector space. Introducing notions of weight and defect relative to an F_q-subspace, we analyze the sequence of maximum…
This dissertation considers new constructions and decoding approaches for error-correcting codes based on non-conventional polynomials, with the objective of providing new coding solutions to the applications mentioned above. With skew…
A matroid base polytope is a polytope in which each vertex has 0,1 coordinates and each edge is parallel to a difference of two coordinate vectors. Matroid base polytopes are described combinatorially by integral submodular functions on a…
We prove that hypersurfaces defined by irreducible square-free polynomials have rational singularities. As an easy consequence, we deduce that certain (possibly non-square-free) polynomials associated to pairs of square-free polynomials…
We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik in 2010. In this paper we consider a hyperplane arrangement associated to every split pseudometric and,…
Following the approach by R. K\"otter and F. R. Kschischang, we study network codes as families of k-dimensional linear subspaces of a vector space F_q^n, q being a prime power and F_q the finite field with q elements. In particular,…
Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic…
We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as…
Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let $q$ be a prime power, $n$ be a positive integer and $\sigma$ be a generator of…
We consider skew free extensions of rings, also known as free multivariate skew polynomial rings, and explore some of the algebraic aspects of this construction. We give different characterizations of such rings and present conditions for…
We relate the notion of matroid pathwidth to the minimum trellis state-complexity (which we term trellis-width) of a linear code, and to the pathwidth of a graph. By reducing from the problem of computing the pathwidth of a graph, we show…
Graphings serve as limit objects for bounded-degree graphs. We define the ``cycle matroid'' of a graphing as a submodular setfunction, with values in [0,1], which generalizes (up to normalization) the cycle matroid of finite graphs. We…
Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble…
For a real polynomial $f$ we present explicit zero-free angular sectors in the complex plane, symmetric with respect to the real axis, with angles depending only on the degree of $f$, and vertices expressed in terms of the coefficients of…
In contrast to matroids, vf-safe delta-matroids have three kinds of minors and are closed under the operations of twist and loop complementation. We show that the delta-matroids representable over GF(4) with respect to the nontrivial…