Raul Cordovil

Sur les Matroïdes Orientés de Rang 3 et les Arrangements de Pseudodroites dans le Plan Projectif Réel

We establish a new geometrical characterization of oriented matroids of rank 3. This characterization is dual to another characterization due to Folkman and Lawrence. We describe the relationship between these characterizations of oriented matroids of rank 3 and the first and second Coxeters projective diagrams of zonohedra.

Cyclic Polytopes and Oriented Matroids

Consider the moment curve in the real euclidean space Rddefined parametrically by the map ?: R?Rd,t???(t) = (t, t2,? , td). The cyclic d -polytopeCd (t1,? , tn) is the convex hull ofn&d different points on this curve. The matroidal analogs are the alternating oriented uniform matroids. A polytope (resp. matroid polytope) is called cyclic if its face lattice is isomorphic to that ofCd (t1,? , tn). We give combinatorial and geometrical characterizations of cyclic (matroid) polytopes. A simple evenness criterion determining the facets ofCd (t1,? , tn) was given by Gale . We characterize the admissible orderings of the vertices of the cyclic polytope, i.e., those linear orderings of the vertices for which Gale?s evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes.

Combinatorial geometries

This special volume is dedicated to the memory of late Michel Las Vergnas who passed away on January 19, 2013 at age 72. Michel Las Vergnaswas a combinatorialmathematicianwith artistic sense and creativity. He is also well known as a cofounder of the theory of oriented matroids which acquired the firm acceptance as fundamental concept in mathematics. Michel was one of the founding editors of European Journal of Combinatorics. On personal side,Michel had broad and deep interest inmusic, arts and culture.Michel was a warm-hearted individual caring for his family and friends with kind and genuine smiles. Michel was the first doctoral student of another great combinatorialist Claude Berge. They were two of the leaders of the Paris school of combinatorics (L’Equipe Combinatoire, Université Paris VI). They influenced greatly the advances of combinatorial mathematics in the world and guided many excellent graduate students. Both Claude and Michel had unusual talents to formulate beautiful conjectures. Throughout his whole career as Directeur de Recherche au CNRS and professor of Université Paris VI (Pierre et Marie Curie), Michel’s devotion to combinatorial mathematics and to the guidance of doctoral students was extraordinary. The late Yahya Ould Hamidoune and the first editor Raul were the first doctoral students ofMichel, graduated in 1978 and 1979, respectively. The third editor Emeric was the last student who completed his study in 2002. Michel had supervised 15 doctoral students. He won the silver medal of the CNRS in 1985. On his pioneering research front, Michel gave many equivalent axiomatizations of oriented matroids in an extensive manuscript,1 which unfortunately has never been fully published. Michel’s theorem on single-element extensions2 of oriented matroids turned out to be crucial in constructing fascinating examples and in resolving degeneracy in the abstract combinatorial setting of optimization. Michel had strong interest in a wide range of combinatorics beyond the oriented

A Commutative Algebra for Oriented Matroids

AbstractLet V be a vector space of dimension d over a field K and let A be a central arrangement of hyperplanes in V. To answer a question posed by K. Aomoto, P. Orlik and H. Terao construct a commutative K -algebra U(A) in terms of the equations for the hyperplanes of A. In the course of their work the following question naturally occurred: \circ Is U(A) determined by the intersection lattice L(A) of the hyperplanes of A? We give a negative answer to this question. The theory of oriented matroids gives rise to a combinatorial analogue of the algebra of Orlik—Terao, which is the main tool of our proofs.

Oriented Matroids and Combinatorial Manifolds

An oriented matroid lattice is a lattice arising from the span of cocircuits of an oriented matroid ordered by conformal relation. One important subclass of the o.m. lattices is the polars of face lattices of zonotopes. In this paper we show that every o.m. lattice is a (combinatorial) manifold. This brings out several interesting results on graphs associated with an o.m. lattice. For example, through Barnettes theorem on connectivity of manifolds, we obtain the (r - 1)-connectivity of the graph of the Las Vergnas lattice and its polar, where r is the rank of the oriented matroid. Furthermore, we prove that the graph of an o.m. lattice is 2(r - 1)-connected, while the graph of its polar is only r-connected. These results are the best possible in the sense that each claimed connectivity is exact for some oriented matroid of rank r. Finally, we give an algorithmic proof of the Bjorner-Edelman-Ziegler theorem: that an oriented matroid is determined by the cograph of the associated o.m. lattice.

Clutters and matroids

Abstract A map on clutters (collections of incomparable sets of a given set) is a function defined from the class of all clutters to itself, that sends a clutter on a ground set E to a clutter on the same set. Here we study two maps on clutters, the blocker map and the complementary map. Our main results include simple characterizations of these maps, which essentially say: the blocker map (the complementary map) is the only nontrivial map interchanging contraction and deletion operations. We also give new forbidden minor characterizations of matroids.

An axiomatic of non-Radon partitions of oriented matroids

Let M ( E , O ) be an oriented matroid. We say that {A, E\A} is a non-Radon partition of M if O A ¯ = O E \ A ¯ is an acyclic reorientation of O . This definition generalizes the classic notion of (non)-Radon partition of a finite subset E of ℝd. We give an intrinsic characterization of the families of partitions which are the family of all non-Radon partitions of some oriented matroid.

Euler's relation, möbius functions and matroid identities

We give a short combinatorial proof of the Euler relation for convex polytopes in the context of oriented matroids. Using counting arguments we derive from the Euler relation several identities holding in the lattice of flats of an oriented matroid. These identities are proven for any matroid by Möbius inversion.

The directions determined by n points in the plane: a matroidal generalization

P.R. Scott posed the problem of determining the minimum number of directions determined by n points which are not all collinear in the plane. We consider a generalization of this problem for oriented matroids. We prove the following theorem: Let M denote an oriented matroid of rank 3. Suppose M has a modular line L, such that the n points of M not in L are not all collinear. Then L has at least 12(n+3) points.

Sur L'Évaluation t(M; 2, 0) du Polynôme de Tutte d'un Matroïde et une Conjecture de B. Grünbaum Relative aux Arrangements de Droites du Plan

Let M be a matroid and let t ( M ; ξ, η) be the Tutte polynomial of M . The lower and upper bound of t ( M ; 2, 0) is calculated. We also prove completely the following B. Grunbaums conjecture, proved for n ⩾ 40 by Purdy [11]: “If A is a non-trivial arrangement of n lines in the real projective plane ( n ⩾ 9) and if f 2 ( A ), the number of regions in the arrangement, is greater than 3 n - 5, then f 2 ( A ) ⩾ 4 n -12.”