Related papers: Characteristic Sets of Matroids
We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results we develop a…
In this paper, the correspondence between the finite dimensional representations of a simple Lie algebra and their characteristic polynomials is established, and a monoid structure on these characteristic polynomials is constructed.…
In this note, we embed the set of all Fricke characters of a free group F -- the set of all characters of representations of F into SL(2,C) -- as an irreducible affine variety V in complex affine space of dimension 2^n-1. Using the Horowitz…
We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for…
We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1, then…
Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as…
Let ${\mathcal{M} = (M_i \colon i\in K)}$ be a finite or infinite family consisting of matroids on a common ground set $E$ each of which may be finitary or cofinitary. We prove the following Cantor-Bernstein-type result: If there is a…
In previous work, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral…
A finite unit norm tight frame is a collection of $r$ vectors in $\mathbb{R}^n$ that generalizes the notion of orthonormal bases. The affine finite unit norm tight frame variety is the Zariski closure of the set of finite unit norm tight…
We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the…
In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each…
We classify Frobenius forms, a special class of homogeneous polynomials in characteristic $p>0$, in up to five variables over an algebraically closed field. We also point out some of the similarities with quadratic forms.
We consider the Frobenius algebra of functions on the critical set of the master function of a weighted arrangement of hyperplanes in $\C^k$ with normal crossings. We construct two potential functions (of first and second kind) of variables…
There are various results in the literature which are part of the general philosophy that a finite group for which a certain parameter (for example, the number of conjugacy classes or the maximum number of elements inverted, squared or…
We describe the classification of ranked definably quasi-Frobenius groups of odd type : dihedral configurations are isomorphic to PGL(2, K) for K an algebraically closed field of characteristic other than two; Frobenius groups are spilt and…
The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general…
A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a preorder on these templates and determine the nontrivial…
Let F_o be a non-archimedean locally compact field of residual characteristic not 2. Let G be a classical group over F_o (with no quaternionic algebra involved) which is not of type A_n for n>1. Let b be an element of the Lie algebra g of G…
We show for each positive integer $a$ that, if $\cM$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $n$ such that either every rank-$r$ matroid in $\cM$ can be covered by…