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\markright{\sc the electronic journal of combinatorics 3 (1996), DS\#4\hfill} 
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\def\sign{\mbox{\rm sign}}
\def\pmz{\{{+},{-},0\}}
\def\R{\mbox{$\mathbb R$}}
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\author{{\Large G\"unter M.~Ziegler\thanks{Supported by a DFG
Gerhard-Hess-Forschungsf\"orderungspreis}}\\[2mm]
Department of~Mathematics, MA~7-1\\
Technische Universit\"at Berlin\\
Strasse des 17.~Juni 136\\
10623 Berlin, Germany\\[1truemm]
{\tt ziegler@math.tu-berlin.de}\\
{\tt http://www.math.tu-berlin.de/$\sim$ziegler}\\[4truemm]}

\title{{\LARGE\bf Oriented Matroids Today}}

\date{\small Submitted: October 6, 1995; Accepted: March 26, 1996;
Version 3 of September 10, 1998.\\[2mm]
Mathematics Subject Classification: 52-00 (52B05, 52B30, 52B35, 52B40)}


\maketitle

\begin{abstract}
This {\em dynamic survey} offers an ``entry point''
for current research in oriented matroids. For this, it
provides updates on the 1993 monograph ``Oriented Matroids''
by Bj\"orner, Las~Vergnas, Sturmfels, White \& Ziegler~\cite{BLSWZ},
in three parts:
\begin{enumerate}
\item a sketch of a few ``Frontiers of Research'' in oriented matroid theory,
\item an update of corrections, comments and progress as compared 
        to~\cite{BLSWZ}, and
\item an extensive, complete and up-to-date bibliography of oriented matroids,
        comprising and extending the bibliography of~\cite{BLSWZ}. 
\end{enumerate}
\end{abstract}


\section{Introduction(s).}

Oriented matroids are both important and interesting objects of
study in Combinatorial Geometry, and indispensable tools of
increasing importance and applicability for many other parts of
Mathematics.  The main parts of the theory and some applications were, in
1993, compiled in the quite comprehensive monograph
by Bj\"orner, Las Vergnas, Sturmfels, White \& Ziegler \cite{BLSWZ}.
For other (shorter) introductions and surveys, 
see  
Bachem \& Kern \cite{BachemKern3},
Bokowski \& Sturmfels \cite{BokowskiSturmfels7},
Bokowski \cite{Bokowski-handbook},
Goodman \& Pollack \cite{GP19},
Ziegler \cite[Chapters 6 and 7]{Z-poly},
and, most recently, Richter-Gebert \& Ziegler \cite{RGZ-handbook}.
\bigskip

\noindent
This {\em dynamic survey} provides three parts:
\begin{enumerate}
\itemsep=-1pt
\item a sketch of a few ``Frontiers of Research'' in oriented matroid theory,
\item an update of corrections, comments and progress as compared 
        to~\cite{BLSWZ}, and
\item an extensive, complete and up-to-date bibliography of oriented matroids,
        comprising and extending the bibliography of~\cite{BLSWZ}. 
\end{enumerate}

\section{What is an Oriented Matroid?}

Let $V=(v_1,v_2,\ldots,v_n)$ be a finite, spanning, sequence of vectors
in~$\R^r$, that is, a finite {\em vector configuration}.  With this
vector configuration, one can associate the following sets of data,
each of them encoding the {\em combinatorial structure} of~$V$.
\begin{itemize}
\itemsep=-1pt
\item 
The {\em chirotope} of~$V$ is the map
\begin{eqnarray*}
\chi_{_V}: \{1,2,\ldots,n\}^r & \longrightarrow & \pmz \\
(i_1,i_2,\ldots,i_r) & \longmapsto &\sign(\det(v_{i_1},v_{i_2},\ldots,v_{i_r}))
\end{eqnarray*}
that records for each $r$-tuple of the vectors whether it forms a
positively oriented basis of~$\R^r$, a basis with negative orientation,
or not a basis.
\item
The set of {\em covectors} of~$V$ is
\[
{\cal V^*}(V)\ \ :=\ \ 
\big \{\big(\sign(a^tv_1),\ldots,\sign(a^tv_n)\big)\in\pmz^n: a\in\R^n\big\},
\]
that is, the set of all partitions of $V$ (into three parts)
induced by hyperplanes through the origin.
\item
The collection of {\em cocircuits} of~$V$ is the set 
\[\begin{array}{rcr}
{\cal C}^*(V) \ := \ 
\big\{ \big(\sign(a^tv_1),\ldots,\sign(a^tv_n)\big)\in\pmz^n:
a\in\R^n \mbox{ is orthogonal to a hyperplane~~}\\
 \mbox{ spanned by vectors in~$V$}\big\},
\end{array}\]
of all partitions by ``special'' hyperplanes that are 
spanned by vectors of the configuration~$V$.
\item
The set of {\em vectors} of~$V$ is
\[\begin{array}{rcr}
{\cal V}(V) & := & 
\big\{ \big(\sign(\lambda_1),\ldots,\sign(\lambda_n)\big)\in\pmz^n:
\lambda_1 v_1 + \ldots + \lambda_n v_n = 0\mbox{~~is a linear~~~}\\
&& \mbox{ dependence between vectors in~$V$}\big\}. 
\end{array}\]
\item
The set of {\em circuits} is
\[\begin{array}{rcr}
{\cal C}(V) & := & 
\big\{ \big(\sign(\lambda_1),\ldots,\sign(\lambda_n)\big)\in\pmz^n:
\lambda_1 v_1+\ldots+\lambda_n v_n=0\mbox{~~is a {\em minimal}\qquad~}\\
&& \mbox{ linear dependence between vectors in~$V$}\big\}. 
\end{array}\]
\end{itemize}

\noindent
A simple, but basic, result now states that all of these sets of data
are equivalent, except for a global sign change that identifies $\chi$
with $-\chi$.  Thus whenever one of the data
\[
\{\chi_{_V},-\chi_{_V}\},\qquad
{\cal V}^*(V), \qquad
{\cal C}^*(V), \qquad
{\cal V}(V), \qquad \mbox{\rm\ or } \quad
{\cal C}(V)
\]
is given, one can from this uniquely reconstruct all the others.
\medskip

Furthermore, one has {\bf axiom systems} (see \cite[Chap.~3]{BLSWZ})
for {\em chirotopes}, {\em covectors}, {\em cocircuits}, {\em vectors}
and {\em circuits} that are easily seen to be satisfied by the
corresponding collections above. Thus there are combinatorial
structures, called {\bf oriented matroids}, that can equivalently be
given by any of these five different sets of data, and
defined/characterized in terms of any of the five corresponding axiom
systems.  (The proofs for the equivalences between these data sets
resp.\ axiom systems are not simple.)
\medskip

Vector configurations as discussed above give rise to
oriented matroids of {\em rank} $r$ on $n$ {\em elements} 
(or: on a {\em ground set} of size~$n$).
Usually the ground set is identified with $E=\{1,2,\ldots,n\}$.

Equivalent to vector configurations, one has the model of
(real, linear, essential, oriented) {\em hyperplane arrangements}: 
finite collections
${\cal A}:= (H_1,H_2,\ldots,H_n)$ of hyperplanes 
(linear subspaces of codimension one) in $\R^r$, with the
extra requirement that $H_1\cap \ldots\cap H_n=\{0\}$,
and with a choice of a positive halfspace $H_i^+$ for each
of the hyperplanes. In fact, every vector configuration gives rise to such
an arrangement via $H_i^+:=\{x\in\R^r:  v_i^t x \ge 0\}$,
and from an oriented hyperplane arrangement we recover a vector
configuration by taking the positive unit normals.

More specialized, one has the model of {\em directed graphs}:
if $D=(V,A)$ is a finite directed graph (with 
vertex set $V=\{0,1,2,\ldots,r\}$ and arc set 
$A=\{a_1,\ldots,a_n\} \subseteq V^2$), 
then one has the obvious ``directed circuits'' in the digraph that give
rise to circuits in the sense of sign vectors 
in~${\cal C}(V)\subseteq\pmz^n$, while directed
cuts give rise to covectors, and minimal directed cuts 
give rise to cocircuits. Thus one obtains the oriented matroid
of a digraph, which can also, equivalently, be constructed by
associating with each arc $(i,j)$ the vector $e_i-e_j\in\R^r$,
where we take $e_i$ to be the $i$-th coordinate vector in $\R^r$ for
$i\ge1$, and $e_0:=0$.
 


Although the axiom systems of oriented matroids 
describe the data arising from vector configurations very well, it
is not true that every oriented matroid corresponds to 
a real vector configuration. In other words, there are 
oriented matroids that are not {\em realizable}.
This points to basic theorems and problems in Oriented Matroid Theory:
 
\begin{itemize}
\item
The {\em Topological Representation Theorem}
(see \cite[Chap.~5]{BLSWZ}) shows that while real vector configurations
can equivalently be represented by {\em oriented arrangements of hyperplanes},
general oriented matroids can be represented by 
{\em oriented arrangements of pseudo-hyperplanes}.
\item
There is no finite set of axioms that would characterize the
oriented matroids that are representable by vector configurations.
In fact, even for $r=3$ there are oriented
matroids on $n$ elements that are {\em minimally non-realizable}
for arbitrarily large~$n$. 
\item
The {\em realization problem} is a difficult algorithmic task:
for a given oriented matroid, to decide whether it is realizable, and
possibly find a realization. This statement is a by-product of the
constructions for the Universality Theorem for oriented matroids,
see below.
\end{itemize}


\section{Some Frontiers of Research.}

Currently there is substantial research done on a variety of
aspects and questions; among them are several deep problems of
oriented matroid theory that were thought to be both hard and
fundamental, and are now gradually turning out to be just that.

Here I give short sketches and a few pointers to the
(recent) literature, for just a few selected topics.
(By construction, the selection is very much biased. I plan
to expand and update regularly. Your help and comments are
essential for~that.) 

\subsection{Realization spaces.}

Mn\"ev's Universality Theorem of 1988 \cite{Mnev-universal}
states that every primary semialgebraic
set defined over $\Z$ is  ``stably equivalent''
to the realization space of some oriented matroid of rank~$3$.
In other words, 
the semialgebraic sets of the form 
\[
{\cal R}(X)\ \ :=\ \ \{ Y\in\R^{3\times n}:
\mbox{\rm sign}(\det(X_{i,j,k}))=\mbox{\rm sign}(\det(Y_{i,j,k}))
\mbox{\rm~for all~} 1\le i<j<k\le n\},
\]
for real matrices $X\in\R^{3\times n}$, can be arbitrarily complicated,
both in their topological and their arithmetic properties.
Mn\"ev's even stronger Universal Partition Theorem \cite{Mnev-partition}
announced in 1991 says that essentially every semialgebraic family
appears in the stratification given by the determinant function on
the $(3\times3)$-minors of $(3\times n)$-matrices.

These results are fundamental and far-reaching. For example, via
oriented matroid (Gale) duality they imply universality theorems
for $d$-polytopes with $d{+}4$ vertices.
    
For some time, no complete proofs were available. This has
only recently changed with the complete proof of the Universal Partition
Theorem  by G\"unzel \cite{Gunzel} and by Richter-Gebert \cite{RG-mnev}.
Richter-Gebert \cite[Sect.~2.5]{RG-universal} 
has also --- finally! --- provided a
suitable notion of ``stable equivalence'' of semialgebraic sets
that is {\em weak enough} to make the
Universality Theorems true, and {\em strong enough} to imply
both homotopy equivalence and arithmetic equivalence
(i.e., it preserves the existence of $K$-rational points in the
semialgebraic set for every subfield $K$ of~$\R$).

Further, surprising recent progress is now available with 
Richter-Gebert's \cite{RG-universal,RGZ-universal} 
Universality Theorem (and Universal Partition Theorem) for
$4$-dimensional polytopes, and related to this his
Non-Steinitz theorem for $3$-spheres.
(See \cite{Gunzel2} for a second proof.)

For still another, very recent, interesting universality result, 
concerning the configuration spaces of planar polygons, see 
Kapovich \& Millson \cite{KapovichMillson4}. Kapovich \& Millson
trace the history of their result back to a universality theorem
by Kempe \cite{Kempe} from~1875!

Here are two major challenges that remain in this area: 

\begin{itemize}
\itemsep=-1pt
\item
To construct and understand the smallest oriented matroids
with non-trivial realization spaces.
The smallest {\em known} examples are Suvorov's \cite{Suvorov}
oriented matroid of rank~$3$ on $14$~points with
a disconnected realization space (see also \cite[p.~365]{BLSWZ}), and
Richter-Gebert's \cite{RG-examples}
new example $\Omega^+_{14}$ with the same parameters, which 
additionally has rational realizations, and a non-realizable symmetry.

\item
To provide Universality Theorems for 
{\em simplicial} $4$-dimensional polytopes.
(The Bokowski-Ewald-Kleinschmidt polytope \cite{BokowskiEwaldKleinschmidt} 
is still the only simplicial example known with a non-trivial realization
space; see also Bokowski \& Guedes de Oliveira \cite{BokowskiGdO1}.)
\end{itemize}

\def\MacP{\mathop{\rm MacP}}\def\Gg{{\cal G}}\def\Ee{{\cal E}}
\subsection{Extension spaces, combinatorial Grassmannians, and matroid
  bundles}

{\em Thanks to Laura Anderson for help on this section!}
\medskip

\noindent
The consideration of \emph{spaces of oriented matroids} 
 brings several very different lines of thinking into a common
topological framework. Given a
set $S$ of oriented matroids, we obtain a partial order on $S$ by weak 
maps, and from this we obtain a topological space by taking the order complex
(the simplicial complex given by chains in the partial order; see 
Bj\"orner \cite{Bjorner6}).
This simplicial complex can be viewed as a combinatorial analog
to a vector bundle.
Just as a vector bundle
represents a continuous parametrization of a set of vector spaces, 
this topological space can be viewed as a ``continuous'' 
parametrization of elements of $S$. Such spaces have arisen in several
contexts:
\begin{itemize}
\item If $S$ is the set of non-trivial single-element extensions of a fixed
oriented matroid $M$, the resulting space is the \emph{extension space}
$\Ee(M)$ of $M$.
\item If $S$ is the set of all rank $r$ oriented matroids on a fixed set
of $n$ elements, this space is the \emph{MacPhersonian} $\MacP(r,n)$.
\item If $S$ is the set of all rank $r$ strong map images of a fixed oriented 
matroid $M$, this space is the \emph{combinatorial Grassmannian} 
$\Gg(r,M)$. (In fact, this example essentially encompasses the
previous two: The extension space $\Ee(M)$ of a 
oriented matroid $M$ is a double cover of $\Gg(r(M)-1,M)$, while 
if $M$ is the unique rank $n$ oriented matroid on 
a fixed set of $n$ elements, then $\Gg(r,M)=\MacP(r,n)$.)
\end{itemize}

Extension spaces are closely related to zonotopal tilings 
(via the Bohne-Dress Theorem) and to oriented matroid programs: 
see Sturmfels \& Ziegler \cite{SturmfelsZiegler}. 
The MacPhersonian and combinatorial Grassmannian arise
in MacPherson's theory of \emph{combinatorial differential manifolds}
and \emph{matroid bundles} \cite{MacPherson} \cite{Anderson4})
in which oriented matroids
serve as combinatorial analogs to real vector spaces. 
\newpage

Among the basic conjectures in the field are:
\begin{itemize} 
\item For a rank $n$ oriented matroid $M^n$, the topology of 
$\Gg(k, M^n)$ should be similar to that of the real 
Grassmannian $G(k,\R^n)$.
In particular, it is known that a realization of an oriented matroid 
determines a canonical homotopy class of maps 
$c:G(k,\R^n)\rightarrow\Gg(k,M^n)$~\cite{AndersonDavis}, 
and it is hoped that $c$ is a 
homotopy equivalence, or at least leads to an isomorphism in 
cohomology with various coefficients.
\item The extension space $\Ee(M^n)$ should have the homotopy type of 
an $(n-1)$-sphere if $M^n$ is realizable.
(This is essentially a special case of the above.)
\end{itemize}

There are substantial grounds for pessimism on both conjectures. 
For instance, 
there are examples of non-realizable $M^n$ such that $\Gg(n-1,M^n)$ 
and $\Ee(M^n)$ are not even connected 
(Mn\"ev \& Richter-Gebert~\cite{MRG}). 
In addition, Mn\"ev's 
Universality Theorem implies that for realizable $M^n$ the inverse 
images under~$c$ of points in $\Gg(r,M^n)$ can have arbitrarily 
complicated topology.
However, substantial progress has been made on the topology of $\Ee(M)$
for realizable $M$ \cite{SturmfelsZiegler} \cite{MRG} 
and on the topology of $\Gg(k,M)$ under 
various conditions: for small values of $k$ (Babson~\cite{Babson}), 
for the first few homotopy groups of the MacPhersonian 
(Anderson~\cite{Anderson3}), and for mod~2 cohomology 
(Anderson \& Davis~\cite{AndersonDavis}).
Three related survey articles are Mn\"ev \& Ziegler~\cite{MZ}, 
Anderson~\cite{Anderson4}, and Reiner~\cite{Reiner}. 

The analogy between 
oriented matroids and real vector bundles 
leads to an intriguing and useful interplay between
topology and combinatorics.
On the one hand, appropriate combinatorial adaptations of 
classical topological methods for real vector bundles prove that for 
realizable $M^n$ the map $c:G(k,\R^n)\rightarrow\Gg(k,M^n)$ induces split
surjections in mod~$2$ 
cohomology \cite{AndersonDavis}). 
On the other hand, combinatorial methods can be applied to
topology as well. Any real vector bundle over a triangulated
base space can be ``combinatorialized'' into a matroid bundle
\cite{MacPherson} \cite{AndersonDavis},
giving a combinatorial approach to the study of bundles.
The most notable success in this direction has been
Gel'fand \& MacPherson's \cite{GMP2}
combinatorial formula for the rational Pontrjagin classes
of a differential manifold.


The topological problems discussed in this section
have close connections to classical problems
of oriented matroid theory, such as the following: 
Las Vergnas' conjectures that every
oriented matroid has at least one mutation (simplicial tope)
and that the set of uniform oriented matroids of rank~$r$ on a
given finite set is connected under performing mutations.
In fact, if these conjectures are false, then  
the ``top level'' of the MacPhersonian,
given by all oriented matroids without circuits of size smaller
than $r$ and at most one circuit of size $r$, cannot be connected.
As for the Las Vergnas conjecture, Bokowski \cite{Bokowski6} and
Richter-Gebert \cite{RG5} have the strongest positive resp.\ negative 
partial results; more work is neccessary.

Further work also remains in the understanding of weak and strong maps
--- currently the only comprehensive source is \cite[Section 7.7]{BLSWZ}.
One still has to derive structural information
from the failure of Las Vergnas' strong map factorization conjecture
(disproved by Richter-Gebert in \cite{RG5}) and derive criteria for
situations where factorization is possible.

\subsection{Affine and infinite oriented matroids.}

The Bohne-Dress Theorem, announced by Andreas Dress at the 1989
``Combinatorics and Geometry'' Conference in Stockholm, 
provides a bijection between the 
zonotopal tilings of a fixed $d$-dimensional zonotope~$Z$
and the single-element liftings of the realizable oriented matroid 
associated with~$Z$.
This theorem turned out to be, at the same time,
\begin{itemize}
\itemsep=-1pt
\item fundamental (see e.~g.\ the connection to extension
        spaces of oriented matroids~\cite{SturmfelsZiegler}),
\item  ``intuitively obvious'' (just draw pictures!), and 
\item surprisingly hard to prove; see Bohne~\cite{Bohne1}
        and Richter-Gebert \& Ziegler \cite{RGZ-bohnedress}.
\end{itemize}
Just recently, however, a new and substantially different proof 
of the Bohne-Dress theorem has
become available, by Huber, Rambau \& Santos \cite{HuberRambauSantos}. 
In particular, there are bijections
\[
\left\{\!\begin{tabular}{c}
zonotopal tilings of\\
the zonotope ${\cal Z}(A)$
\end{tabular}\!\right\}\ \ \longleftrightarrow\ \
\left\{\!\begin{tabular}{c}
subdivisions of the\\
Lawrence polytope $\Lambda(A)$
\end{tabular}\!\right\}\ \ \longleftrightarrow\ \
\left\{\!\begin{tabular}{c}
extensions of the dual\\
oriented matroid ${\cal M}^*(A)$
\end{tabular}\!\right\}
\]
The first bijection follows from the ``Cayley trick'',
see Huber, Rambau \& Santos \cite{HuberRambauSantos}.
The second, more difficult one was already before 
established by Santos \cite{Santos2,HuberRambauSantos}.

A separate, simpler proof for the case of rank~3 ---
pseudoline arrangements are in bijection with zonotopal tilings
of a centrally symmetric $2n$-gon --- is contained in the
work by Felsner \& Weil \cite{FelsnerWeil}.
\smallskip

The Bohne-Dress theorem provides a connection 
to several other areas of study.
On the one hand, the classification and enumeration 
of rhombic tilings of a hexagon relates to the theory of
plane partitions and symmetric functions;
see e.g.\ Elnitzky \cite{Elnitzky}, 
Edelman \& Reiner \cite{EdelmanReiner1}.

On the other hand, there is a definite need for a better understanding of
zonotopal tilings of the entire plane (or of $\R^d$).
Two different approaches have been started 
by Bohne \cite[Kapitel~5]{Bohne2}
and by Crapo \& Senechal~\cite{CrapoSenechal}, but no complete
picture has emerged, yet.
This is of interest, for example, in view
of the mathematical problems posed by understanding quasiperiodic tilings
and quasicrystals; see Senechal~\cite{Senechal,Senechal2}.


\subsection{Realization algorithms.}

The realizability problem --- given a ``small'' oriented matroid,
find a realization or prove that none exists --- is a 
key problem not only in oriented matroid
theory, but also for various applications, such
as the classification of ``small'' simplicial spheres into polytopal
and non-polytopal ones (see 
Bokowski \& Sturmfels~\cite{BokowskiSturmfels5,BokowskiSturmfels7}, 
Altshuler, Bokowski \& Steinberg \cite{ABSt},
Bokowski \& Shemer \cite{BokowskiShemer}). 
The universality theorems mentioned above tell
us that the problem is hard: in fact, 
in terms of Complexity Theory is just as hard as the 
``Existential Theory of the Reals,'' the problem of solving 
general systems of algebraic equations and inequalities over the
reals. While it is not known whether the problem over~$\Q$ is at all
algorithmically solvable (see Sturmfels~\cite{Sturmfels7}), 
there are algorithms available that
(at least theoretically) solve the problem over the reals.
For the general problem  
Basu, Pollack \& Roy~\cite{BasuPollackRoy2} currently have the
best result:
\begin{itemize}
\item[]
Let  ${\cal P}=\{P_1,\ldots,P_s\}$  be a set of polynomials in $k<s$ 
variables each of degree at most $d$ and each with coefficients in a
subfield $K\subseteq\R$.\\
There is an algorithm which finds a solution in each connected
component of the solution set, for each 
sign condition on $P_1,\ldots,P_s$, in at most 
${O(s)\choose k} s\, d^{O(k)}=(s/k)^k s\, d^{O(k)}$ 
arithmetic operations in~$K$.
\end{itemize}
However, until now this is mostly of theoretical value. 
What can be done for specific, explicit, small examples?
Given an oriented matroid of rank~3, it seems that 
\begin{itemize}
\itemsep=-1pt
\item the most efficient algorithm (in practice) currently 
     available to {\em find a realization} (if one exists)
     is the iterative ``rubber band'' algorithm described in
     Pock~\cite{Pock}.
\item the most efficient algorithm (in practice) currently 
     available to {\em show that it is not realizable} (if it isn't)
     is the ``binomial final polynomials'' algorithm of 
     Bokowski \& Richter-Gebert \cite{BokowskiRichter} which uses
     solutions of an auxiliary linear program to construct final polynomials.
     (An explicit example of a non-realizable oriented matroid
     $\Omega^-_{14}$ without a biquadratic final polynomial was just recently
     constructed by Richter-Gebert \cite{RG-examples}.)
\end{itemize}
Neither of these two parts is guaranteed to work: but still the
combination of both parts was good enough for a (still unpublished)
complete classification of all $312{,}356$ (unlabeled reorientation
classes of) uniform oriented matroids of rank~$3$ on~$10$ points into
realizable and non-realizable ones (Bokowski, Laffaille \&
Richter-Gebert \cite{BokowskiLaffailleRichterG}).

A very closely related topic is that of Automatic Theorem Proving in
(plane) geometry.  In fact, the question of validity of a certain
incidence theorem can be viewed as the realizability problem for
(oriented or unoriented) matroids of the configuration.
Richter-Gebert's Thesis \cite{RG3} and Wu's book \cite{Wu} here present
two recent (distinct) views of the topic, both with many of its
ramifications.

Here we are far from having reached the full scope of current
possibilites.  For an (impressive) demonstration I refer to the
spectacular new 
Interactive Geometry Software system {\sc Cinderella} by 
Ulrich Kortenkamp and J\"urgen Richter-Gebert
\cite{Cinderella}, whose prover includes the idea of ``binomial proofs''
\cite{CrapoRG1} as well as new randomized methods.
An amazing piece of work!


\section{Some Additions and Corrections.}

In this section, I collect some notes, additions, corrections and
updates to the 1993 book by Bj\"orner, Las Vergnas, Sturmfels, White \&
Ziegler \cite{BLSWZ}.  The list is far from complete (even in view of
the points that I know about), and with your help I plan to expand it
in the future.

\subsubsection*{Page 150, Section 3.9 ``Historical Sketch''}
Jaritz \cite{Jaritz1,Jaritz2} gives a new axiomatic of
oriented matroids in terms of ``order functions'' whose axioms
and concepts she traces back to Sperner \cite{Sperner} (1949!),
Karzel \cite{Karzel} etc.
At the same time, Kalhoff \cite{Kalhoff} reduces embedding questions
about pseudoline arrangements, as solved by Goodman, Pollack, Wenger
\& Zamfirescu \cite{GPWengerZ2,GPWengerZ5}, back to 1967 
results of Prie\ss-Crampe~\cite{Priess-Crampe}.

All this gets us closer to confirming the suspicion that probably
Hilbert knew about oriented matroids\ldots

\subsubsection*{Page 220, Exercise 4.28\boma{^*}.}
Part (a) of this was already proved by Zaslavsky \cite[Sect.~9]{Zaslavsky1}.
However, part (b) remains open and should be an interesting challenge.

\subsubsection*{Page 227, Definition 5.1.3.}
For condition (A2), if
$S_A\cap S_e=S^{-1}=\emptyset$ is the empty sphere 
in a zero sphere $S_A\cong S^0$, then the sides of this empty sphere are
the two points of $S_A$.

\subsubsection*{Page 244, Exercise 5.2(c).}
Hochst\"attler \cite{Hochstattler7} has shown that quite general
arrangements of wild spheres also yield oriented matroids.

\subsubsection*{Page 270, Proposition~6.5.1.}
Felsner \cite{Felsner} has constructed a new and especially 
effective encoding scheme for wiring diagrams, which implies
improved upper bound for the number of wiring diagrams and hence
of simple pseudoline arrangements, namely
\[
\log_2 s_n\ <\ 0.6988\,n^2.
\]

\subsubsection*{Page 275:}

Richter-Gebert~\cite{RG-orientability} has proved (in 1996, and 
written up in 1998) that 
orientability is NP-complete \cite{RG-orientability}.
(It's a beautiful paper!)
 
\subsubsection*{Page 279, Exercises 6.21(a)\boma{^{(*)}}}
The answer is ``yes'': this problem was solved in 1997,
with an explicit construction, by
Forge \& Ram\'{\i}rez Alfons\'{\i}n \cite{ForgeRAlfonsin}.
 
\subsubsection*{Page 334, Exercises 7.15(b)\boma{^{(*)}} and 7.17.}
An explicit example of an oriented matroid that has a simple adjoint, but not
a double adjoint was constructed by Hochst\"attler \& Kromberg
\cite{HochstattlerKromberg1,Kromberg}.

Also, they observed \cite{HochstattlerKromberg0,Kromberg} that some assertions
in Exercise 7.17 are not correct:  J\"urgen Richter-Gebert's
\cite[p.~117]{RG3} 8-point torus is realizable over an ordered {\em
skew field}, but not over~$\R$.  Therefore the oriented matroid given
by such a skew realization has an infinite sequence of adjoints, but it
is not realizable in~$\R^4$.

\subsubsection*{Page 337, Exercises 7.44*.}

No one seems to remember the example: so consider this to be an
open problem.
(The non-existence of such an example is also discussed, as a Conjecture of
Brylawski, in McNulty~\cite{McNulty2}.) 

\subsubsection*{Page 385, McMullen's problem on projective transformations.}

Forge \& Schuchert \cite{ForgeSchuchert} have found a configuration
of $10$ points in general position in affine $4$-space that no  
projective transformation can put into convex position.
This solves McMullen's problem for $d=4$ resp.\ $r=5$ with 
$f(4)=g(5)=9$.

This is consistent with the conjecture that
the inequalities  $2d+1\le g(d+1)\le f(d)$ [sic.!]
hold with equality also for $d>4$.

\subsubsection*{Page 396.}

Proposition 9.4.2 is true only for $n\ge r+2$. For
$n=r+1$ the matroid is one single circuit, the
inseparability graph is a complete graph, etc.
 
\subsubsection*{Page 405 (top).}

It is not true that the sphere ${\cal S}={M}^9_{963}$ is neighborly:
the edges 13 and 24 are missing (in the labeling used in
\cite{BLSWZ}).  Thus Shemer's Theorem~9.4.13 cannot be applied here. A
proof that the sphere admits at most one matroid polytope, 
{\tt AB}$(9)$, was given by Bokowski \cite{Bokowski1} in 1978
(see also Altshuler, Bokowski \& Steinberg \cite{ABSt}  
and Antonin \cite{Antonin}). It is described in detail
in Bokowski \& Schuchert \cite{BokowskiSchuchert1}.
(The oriented matroid {\tt RS}$(8)$ discussed in \cite[Sect.~1.5]{BLSWZ} 
arises as a contraction of the oriented matroid {\tt AB}$(9)$.)

\subsubsection*{Page 413, Exercise 9.12\boma{^{(*)}}.}

Bokowski \& Schuchert \cite{BokowskiSchuchert1} showed that the
smallest example (both in terms of its rank $r=5$ and in terms of its
number of vertices $n=9$), is given by Altshuler's sphere $M^9_{963}$.

\subsubsection*{Page 424.}

In Definition 10.1.8, delete 
``infeasible oriented matroid program'' resp.\
``unbounded  oriented matroid program.''

After this, 
the cocircuit $Y  $ should be $Y  =(0 0 {+}{+}{+}{|}{\bf{+}{-}})$,
the   circuit $X  $ should be $X  =(0{+} 0  0 {+}{|}{\bf{-}{+}})$, and 
the   circuit $X_0$ should be $X^0=(0 0  0 {+}{+}{|}{\bf{-}{+}})$

\subsubsection*{Page 426, Proof of Corollary 10.1.10.}

``Orthogonality of circuits and {\bf co}circuits''


\newpage

\section{The Bibliography.}

The purpose of the following is to keep the bibliography of the book 
\cite{BLSWZ}
up-to-date electronically. For this, the following contains {\em all\/} the
references of this book (including those which are not directly concerned with
oriented matroids). Into this I have inserted all the 
corrections, missing references, 
additions and updates that I am currently aware of.
Any corrections, new papers concerned with oriented matroids, and
other updates that you tell me about will be entered asap.
I am eager to hear about your corrections, updates and comments!

Related bibliographies on the web are:
\begin{itemize}
\item
Bibliography of signed and gain graphs, by Thomas Zaslavsky,
published as a dynamic survey DS8 in the
{\sl the electronic journal of combinatorics} {\bf 3} (1996), DS\#4;
published July 20, 1998,\\
{\tt http://www.combinatorics.org/Surveys/index.html}
%%  {\tt http://math.binghamton.edu/zaslav/}
%%        Subject: Bibliography_of_Signed_and_Gain_Graphs
\item
Bibliography of matroids, by Sandra Kingan, at \\
{\tt http://members.aol.com/matroids/biblio.htm}
\end{itemize}

\renewcommand{\ref}[3]{{\sc#1} \it #2 \rm #3}
\begin{thebibliography}{888}
\itemsep=-2.1pt

\bibitem{Aigner}\ref
{M.~Aigner:} 
{Combinatorial Theory,}
{Grundlehren Series {\bf 234}, Springer 1979.}
 
\bibitem{AlfterHochstaettler}\ref
{M.~Alfter \& W.~Hochst\"attler:}
{On pseudomodular matroids and adjoints,}
{{\sl Discrete Math.\ Appl.} {\bf 60} (1995), 3-11.}
   
\bibitem{Alfonsin}\ref
{J. L. R. Alfons\'{\i}n:}
{Spatial graphs and oriented matroids: the trefoil,}
{Preprint 1997, 14 pages.}

\bibitem{AlfterKernWanka}\ref
{M.~Alfter, W.~Kern \& A.~Wanka:}
{On adjoints and dual matroids,} 
{{\sl J.~Combinatorial Theory}, Ser.~B {\bf 50} (1990), 208-213.}

\renewcommand\thefootnote{*}
\bibitem{ALV}\ref
{L.~Allys \& M.~Las Vergnas:}
{Minors of matroid morphisms,}
{{\sl J.~Combinatorial Theory}, Ser.~B (1991), to appear 
(?).}\footnote{References with an asterisk do not really seem to exist.}
 
\bibitem{Alon}\ref
{N.~Alon:} 
{The number of polytopes, configurations and real matroids,}
{{\sl Mathematika} {\bf 33} (1986), 62-71.}
  
\bibitem{AlonGyory}\ref
{N.~Alon \& E.~Gy\"ory:} 
{The number of small semispaces of a finite set of points in the plane,}
{{\sl J.~Combinatorial Theory}, Ser.~A {\bf 41} (1986), 154-157.}

\bibitem{Altshuler}\ref
{A.~Altshuler:} 
{Neighborly $4$-polytopes and neighborly 
combinatorial $3$-manifolds with ten vertices,}
{{\sl Canadian J.~Math.} {\bf 29} (1977), 400-420.}

\bibitem{ABS1}\ref
{A.~Altshuler, J.~Bokowski \& P.~Schuchert:}
{Spatial polyhedra without diagonals,}
{{\sl Israel J.\ Math.} {\bf 86} (1994), 373--396.}

\bibitem{ABS2}\ref
{A.~Altshuler, J.~Bokowski \& P.~Schuchert:}
{Sphere systems and neighborly spatial polyhedra with $10$ vertices,}
{in: {\sl First international conference on stochastic
geometry, convex bodies and empirical measures}, Palermo 1993 (M. Stoka, ed.),
{\sl Circolo Matematico di Palermo, Suppl.\ 
Rend.\ Circ.\ Mat.\ Palermo, II.\ Ser.} {\bf 35} (1994), 15-28.}

\bibitem{ABS3}\ref
{A.~Altshuler, J.~Bokowski \& P.~Schuchert:}
{Neighborly $2$-manifolds with $12$ vertices,}
{{\sl J. Combinatorial Theory}, Ser.~A, {\bf 75} (1996), 148-162.}

\bibitem{ABSt}\ref
{A.~Altshuler, J.~Bokowski \& L.~Steinberg:}
{The classification of simplicial $3$-spheres with nine vertices
into polytopes and nonpolytopes,}
{{\sl Discrete Math.} {\bf 31} (1980), 115-124.}
 
\bibitem{Anderson1}\ref
{L. Anderson:}
{Topology of Combinatorial Differential Manifolds,}
{Ph.~D.\ Thesis, MIT 1994, 40~pages.}

\bibitem{Anderson2}\ref
{L. Anderson:}
{Topology of combinatorial differential manifolds,}
{Preprint 1995, 29 pages; {\sl Topology}, to appear.}

\bibitem{Anderson3}\ref
{L. Anderson:}
{Homotopy groups of the combinatorial Grassmannians,}
{Preprint, University of Indiana, Bloomington 1996, 15~pages;
{\sl Discrete Comput.\ Geometry}, to appear.}

\bibitem{Anderson4}\ref
{L. Anderson:}
{Matroid bundles,}
{Preprint 1998, 21~pages;
in: ``New Perspectives in Algebraic Combinatorics'',
MSRI Book Series, Cambridge University Press, to appear.}

\bibitem{AndersonWenger}\ref
{L. Anderson \& R. Wenger:}
{Oriented matroids and hyperplane transversals,}
{{\sl Advances Applied Math.} {\bf 119} (1996), 117-125.}

\bibitem{AndersonDavis}\ref
{L. Anderson \& J. F. Davis:}
{Mod~$2$ cohomology of the combinatorial Grassmannian,}
{Preprint 1998.}

\bibitem{Ando}\ref
{T.~Ando:} 
{Totally positive matrices,} 
{{\sl Linear Algebra Appl.} {\bf 90} (1987), 165-219.}
 
\bibitem{Antonin}\ref
{C.~Antonin:}
{Ein Algorithmusansatz f\"ur Realisierungsfragen im $E^d$ getestet
an kombinatorischen $3$-Sph\"aren,}
{Staatsexamensarbeit, Universit\"at Bochum 1982.}

\bibitem{Arnold}\ref
{V.~I.~Arnol'd:}
{The cohomology ring of the colored braid group,}
{{\sl Mathematical Notes} {\bf 5} (1969), 138-140.}
 
\bibitem{AssKlei}\ref
{S.~F.~Assmann \& D.~J.~Kleitman:}
{Characterization of curve map graphs,}
{{\sl Discrete Applied Math.} {\bf 8} (1984), 109-124.}

\bibitem{ADRS}\ref
{C. Athanasiadis, J. De Loera, V. Reiner \& F. Santos:}
{Fiber polytopes for the projections between cyclic polytopes,}
{Preprint, 28 pages; {\sl European J. Combinatorics}, to appear.}

\bibitem{Atiyah}\ref
{M.~F.~Atiyah:}
{Convexity and commuting Hamiltonians,}
{{\sl Bulletin London Math.\ Soc.} {\bf 14} (1982), 1-15.}
 
\bibitem{AvisFukuda1}\ref
{D.~Avis \& K.~Fukuda:}
{A basis enumeration algorithm for linear systems with geometric applications,}
{{\sl Applied Math.\ Letters} {\bf 4} (1991), 39-42.}

\bibitem{AvisFukuda2}\ref
{D.~Avis \& K.~Fukuda:}
{A pivoting algorithm for convex hulls and vertex enumeration of
arrangements and polyhedra,}
{{\sl Discrete Comput.\ Geometry} {\bf 8} (1992), 295-313.}

\bibitem{AvisFukuda3}\ref
{D.~Avis \& K.~Fukuda:}
{Reverse search for enumeration,}
{{\sl Discrete Applied Math.} {\bf 65} (1996), 21-46.} 

\bibitem{AzoalaSantos}\ref
{M. Azoala \& F. Santos:}
{The graph of triangulations of $d+4$ points is $3$-connected,}
{Preprint 1998.}

\bibitem{Babson}\ref
{E.~K.~Babson:}
{A Combinatorial Flag Space,}
{Ph.D.~Thesis, MIT 1992/93, 40~pages.}

\bibitem{Bachem1}\ref
{A.~Bachem:}  
{Convexity and optimization in discrete structures,} 
{in: ``Convexity and Its Applications'' (P.~M.~Gruber, J.~Wills, eds.),
Birkh\"auser, Basel 1983, pp.~9-29.}

\bibitem{Bachem2}\ref
{A.~Bachem:}   
{Polyhedral theory in oriented matroids,} 
{in: {\sl Mathematical Programming}, 
Proc.~Int.\ Congress, Rio de Janeiro 1981, North Holland 1984, pp.~1-12.}
 
\bibitem{BachemDressWenzel}\ref
{A.~Bachem, A.~W.~M.~Dress \& W.~Wenzel:} 
{Five variations on a theme by Gyula Farkas,} 
{{\sl Advances Appl.\ Math.} {\bf 13} (1992), 160-185.}
 
\bibitem{BachemKern1}\ref
{A.~Bachem \& W.~Kern:}   
{Adjoints of oriented matroids,} 
{{\sl Combinatorica} {\bf 6} (1986), 299-308.}
 
\bibitem{BachemKern2}\ref
{A.~Bachem \& W.~Kern:}   
{Extension equivalence of oriented matroids,} 
{{\sl European J.~Combinatorics} {\bf 7} (1986), 193-197.}
  
\bibitem{BachemKern3}\ref
{A.~Bachem \& W.~Kern:}  
{Linear Programming Duality. An Introduction to Oriented Matroids,} 
{Universitext, Springer-Verlag, Berlin 1992.}
 
\bibitem{BachemKern4}\ref
{A.~Bachem \& W.~Kern:}  
{A guided tour through oriented matroid axioms,} 
{{\sl Acta Math.\ Appl.\ Sin.}, Engl.\ Ser.~{\bf 9} (1993), 125-134.}

\bibitem{BachemReinhold}\ref
{A.~Bachem \& A.~Reinhold:}  
{On the complexity of the Farkas property of oriented matroids,} 
{preprint 89.65, Universit\"at K\"oln 1989.}

\bibitem{BachemWanka1}\ref
{A.~Bachem \& A.~Wanka:}   
{On intersection properties of (oriented) matroids,}
{{\sl Methods of Operations Research} {\bf 53} (1985), 227-229.}
 
\bibitem{BachemWanka2}\ref
{A.~Bachem \& A.~Wanka:}   
{Separation theorems for oriented matroids,} 
{{\sl Discrete Math.} {\bf 70} (1988), 303-310.}
 
\bibitem{BachemWanka3}\ref
{A.~Bachem \& A.~Wanka:}   
{Euclidean intersection properties,} 
{{\sl J.~Combinatorial Theory}, Ser.~B {\bf 47} (1989), 10-19.}
 
\bibitem{BachemWanka4}\ref
{A.~Bachem \& A.~Wanka:}  
{Matroids without adjoints,} 
{{\sl Geometriae Dedicata} {\bf 29} (1989), 311-315.}

\bibitem{BaclawskiWhite}\ref
{K.~Bac\l awski \& N.~White:} 
{Higher order independence in matroids,}
{{\sl J.\ London Math.\ Society} {\bf 19} (1979), 193-202.}
 
\bibitem{Balinski}\ref 
{M.~L.~Balinski:}
{On the graph structure of convex polyhedra in $n$-space,}
{{\sl Pacific J.\ Math.} {\bf 11} (1961), 431-434.}
 
\bibitem{Barnette1}\ref 
{D.~W.~Barnette:}  
{Diagrams and Schlegel diagrams,} 
{in: {\sl Combinatorial Structures and their Applications},
Gordon and Breach, New York 1970, pp.~1-4.}

\bibitem{Barnette2}\ref 
{D.~W.~Barnette:} 
{A proof of the lower bound conjecture for convex polytopes,} 
{{\sl Pacific J.~Math.} {\bf 46} (1973), 349-354.}
 
\bibitem{Barnette3}\ref 
{D.~W.~Barnette:} 
{Graph theorems for manifolds,} 
{{\it Israel J.\ Math.} {\bf 16} (1973), 62-72.}
 
\bibitem{Barnette4}\ref
{D.~W.~Barnette:}
{Two ``simple'' $3$-spheres,}
{{\sl Discrete Math.} {\bf 67} (1987), 97-99.}
 
\bibitem{Barvinok}\ref
{A.~I.~Barvinok:} 
{On the topological properties of spaces of polytopes,} 
{in: {\sl Topology and Geometry --- Rohlin Seminar} (O.Ya.~Viro, ed.),
Lecture Notes in Mathematics {\bf 1346}, Springer 1988, pp.~495-500.}
 
\bibitem{BasuPollackRoy}\ref
{S. Basu, R. Pollack \& M.-F. Roy:}  
{On computing a set of points meeting every cell
defined by a family of polynomials on a variety,}
{{\sl J. Complexity} {\bf 13} (1997), 28-37.}

\bibitem{BasuPollackRoy2}\ref
{S. Basu, R. Pollack \& M.-F. Roy:}  
{A new algorithm to find a point in every cell 
defined by a family of polynomials,} 
{in: ``Quantifier Elimination and Cylindric Algebraic Decomposition''
(B. Caviness, J. Johnson, eds.), 
{\sl Texts and Monographs in Symbolic Computation},
Springer-Verlag, Wien, New-York 1997.} 
 
\bibitem{BasuPollackRoy3}\ref
{S. Basu, R. Pollack \& M.-F. Roy:}  
{On the combinatorial and algebraic complexity of
quantifier elimination,}
{{\sl J. ACM} {\bf 43} (1996), 1002-1055.}

\bibitem{BayerLee}\ref
{M.~M.~Bayer, C.~W.~Lee:}
{Combinatorial aspects of convex polytopes,}
{in: {\sl Handbook of Convex Geometry} 
(P.~Gruber, J.~Wills, eds.), North-Holland, Amsterdam 1993, pp.~485--534.}

\bibitem{BayerSturmfels}\ref
{M.~Bayer \& B.~Sturmfels:} 
{Lawrence polytopes,}
{{\sl Canadian J.~Math.} {\bf 42} (1990), 62-79.}
 
\bibitem{Becker}\ref
{E.~Becker:} 
{On the real spectrum of a ring and its applications 
to semialgebraic geometry,}
{{\sl Bulletin Amer.\ Math.\ Soc.} {\bf 15} (1986), 19-60.}

\bibitem{BenedettiRisler}\ref
{R.~Benedetti \& J.-J.~Risler:} 
{Real Algebraic and Semi-algebraic Sets,} 
{Hermann, Paris 1990.}
 
\bibitem{BensonGrove}\ref
{C.~T.~Benson \& L.~C.~Grove:} 
{Finite Reflection Groups (second edition),}
{Springer 1985.}

\bibitem{BFZ}\ref
{A.~Berenstein, S.~Fomin \& A.~Zelevinsky:}
{Parametrizations of canonical bases and totally positive matrices,}
{{\sl Advances in Math.} {\bf 122} (1996), 49-149.}

\bibitem{BergeLasVergnas}\ref
{C.~Berge \& M.~Las Vergnas:}   
{Transversals of circuits and acyclic orientation in graphs and matroids,}
{{\sl Discrete Math.} {\bf 50} (1984), 107-108.} 

\bibitem{BEPY}\ref 
{M.~Bern \& P.~Eppstein \& P.~Plassmann \& F.~Yao:} 
{Horizon theorems for lines and polygons,}
{in: {\sl Discrete and Computational Geometry: Papers from the DIMACS
Special Year} (ed.\ J.~E.~Goodman, R.~Pollack, W.~Steiger),
DIMACS Series in Discrete Mathematics and Theoretical Computer Science,
Vol.~6, Amer.\ Math.\ Soc.\ 1991, pp.~45-66.} 
 
\bibitem{BieniaCordovil}\ref
{W.~Bienia \& R.~Cordovil:}   
{An axiomatic of non-Radon partitions of oriented matroids,} 
{{\sl European J.~Combinatorics} {\bf 8} (1987), 1-4.}
 
\bibitem{BieniaDaSilva}\ref
{W.~Bienia \& I.~P.~da Silva:}   
{On the inversion of one base sign in an oriented matroid,} 
{{\sl J.~Combinatorial Theory} Ser.~B {\bf 90} (1990), 299-308.}
 
\bibitem{BieniaLasVergnas}\ref
{W.~Bienia \& M.~Las Vergnas:}  
{Positive dependence in oriented matroids, }
{preprint 1990.}
 
\bibitem{Bienstock}\ref
{D.~Bienstock:} 
{Some provably hard crossing number problems,}
{{\sl Proc.\ 6th ACM Ann.\ Symp.\ on Computational Geometry} 
(Berkeley, June 1990), ACM 1990, pp.~253-260.}

\bibitem{BilleraBrownDiaconis}\ref
{L. J. Billera, K. S. Brown \& P. Diaconis:}
{Geometry and probability in three dimensions,}
{Preprint, Cornell 1998, 23~pages.}

\bibitem{BER1}\ref
{L. J. Billera, R. Ehrenborg \& M. Readdy:}
{The $cd$-index of zonotopes and arrangements,}
{Preprint, Cornell 1997, 13 pages.}

\bibitem{BER2}\ref
{L. J. Billera, R. Ehrenborg \& M. Readdy:}
{The $c$-$2d$-index of oriented matroids,}
{{\sl J.~Combinatorial Theory, Ser.~A} {\bf 80} (1997), 79-105.}
 
\bibitem{BFS}\ref
{L.~J.~Billera, P.~Filliman \& B.~Sturmfels:} 
{Constructions and complexity of secondary polytopes,}
{{\sl Advances in Math.} {\bf 83} (1990), 155-179.}

\bibitem{BGS}\ref
{L.~J.~Billera, I.~M.~Gel'fand \& B.~Sturmfels:} 
{Duality and minors of secondary polyhedra,}
{{\sl J.~Combinatorial Theory, Ser.~B} {\bf 57} (1993), 258--268.}
 
\bibitem{BKS}\ref
{L.~J.~Billera, M.~M.~Kapranov \& B.~Sturmfels:} 
{Cellular strings on polytopes,}
{{\sl Proc.\ Amer.\ Math.\ Soc.} {\bf 122} (1994), 549--555.}

\bibitem{BilleraLee}\ref 
{L.~J.~Billera \& C.~W.~Lee:} 
{A proof of the sufficiency of McMullen's conditions
for $f$-vectors of simplicial polytopes,}
{{\sl J.~Combinatorial Theory} Ser.~A {\bf 31} (1981), 237-255.} 
 
\bibitem{BilleraMunson1}\ref
{L.~J.~Billera \& B.~S.~Munson:}  
{Polarity and inner products in oriented matroids,}
{{\sl European J.~Combinatorics} {\bf 5} (1984), 293-308.}
 
\bibitem{BilleraMunson2}\ref
{L.~J.~Billera \& B.~S.~Munson:}   
{Oriented matroids and triangulations of convex polytopes,}
{in: {\sl Progress in Combinatorial Optimization} (Proc.~Waterloo Silver
Jubilee Conference 1982), Academic Press, Toronto 1984, 17-37.}
 
\bibitem{BilleraMunson3}\ref
{L.~J.~Billera \& B.~S.~Munson:}   
{Triangulations of oriented matroids and convex polytopes,} 
{{\sl SIAM J.~Algebraic Discrete Methods} {\bf 5} (1984), 515-525.}
 
\bibitem{BilleraSturmfels1}\ref 
{L.~J.~Billera \& B.~Sturmfels:} 
{Fiber polytopes,}
{{\sl Annals of Mathematics} {\bf 135} (1992), 527--549.}
  
\bibitem{BilleraSturmfels2}\ref 
{L.~J.~Billera \& B.~Sturmfels:} 
{Iterated fiber polytopes,}
{{\sl Mathematika} {\bf 41} (1994), 348-363.}

\bibitem{Bing}\ref
{R~H~Bing:} 
{Some aspects of the topology of $3$-manifolds related to the Poincar\'e
conjecture,}
{in: {\sl Lectures on Modern Mathematics, Vol.~II} (T.L.~Saaty, ed.),
Wiley 1964, pp.~93-128.}
 
\bibitem{Bjorner1}\ref
{A.~Bj\"orner:} 
{Shellable and Cohen-Macaulay partially ordered sets,}
{{\sl Transactions Amer.\ Math.\ Soc.} {\bf 260} (1980), 159-183.}

\bibitem{Bjorner2}\ref
{A.~Bj\"orner:} 
{On complements in lattices of finite length,}
{{\sl Discrete Math.} {\bf 36} (1981), 325-326.}
 
\bibitem{Bjorner3}\ref
{A.~Bj\"orner:}
{Posets, regular $CW$ complexes and Bruhat order,}
{{\sl Europ.\ J.\ Combinatorics} {\bf 5} (1984), 7-16.}
 
\bibitem{Bjorner4}\ref
{A.~Bj\"orner:}
{Orderings of Coxeter groups,} 
{in: ``Combinatorics and Algebra'' (C.~Greene, ed.), {\sl Contemporary Math.}
{\bf 34}, Amer.\ Math.\ Soc.\ 1984, 175-195.}
 
\bibitem{Bjorner5}\ref
{A.~Bj\"orner:} 
{Some combinatorial and algebraic properties of 
Coxeter complexes and Tits buildings,}
{{\sl Advances in Math.} (1984) {\bf 52}, 173-212.}
 
\bibitem{Bjorner6}\ref
{A.~Bj\"orner:} 
{Topological methods,}
{in: {\sl Handbook of Combinatorics} 
(eds.\ R.~Graham, M.~Gr\"otschel, L.~Lov\'asz), North-Holland/Elsevier,
Amsterdam 1995, 1819-1872.}

\bibitem{BEZ}\ref
{A.~Bj\"orner, P.~H.~Edelman \& G.~M.~Ziegler:}   
{Hyperplane arrangements with a lattice of regions, }
{{\sl Discrete Comput.~Geometry} {\bf 5} (1990), 263-288.}
  
\bibitem{BjornerKalai}\ref
{A.~Bj\"orner \& G.~Kalai:}
{Extended Euler-Poincar\'e relations for cell complexes,}
{in: {\sl Applied Geometry and Discrete Mathematics -- 
The Victor Klee Festschrift} (P.~Gritzmann, B.~Sturmfels, eds.),
DIMACS Series in Discrete Mathematics and Theoretical Computer Science,
Amer.\ Math.\ Soc.~{\bf 4} (1991), 81-89.}
 
\bibitem{BLSWZ}\ref
{A.~Bj\"orner, M.~Las Vergnas, B.~Sturmfels, N.~White \& G.~M.~Ziegler:}
{Oriented Matroids,}
{{\sl Encyclopedia of Mathematics}, Vol.~46, Cambridge University Press 1993.}
 
\bibitem{BjornerWachs1}\ref
{A.~Bj\"orner \& M.~Wachs:} 
{On lexicographically shellable posets,}
{{\sl Transactions Amer.\ Math.\ Soc.} {\bf 277} (1983), 323-341.}

\bibitem{BZ-shellabs}\ref
{A.~Bj\"orner \& G.~M.~Ziegler:}   
{Shellability of oriented matroids,} 
{Abstract, Workshop on 
{\sl Simplicial Complexes}, Institute for Mathematics and its 
Applications, University of Minnesota, Minneapolis, March 1988; and
Abstract, Conf.\ on {\sl Ordered Sets}, Oberwolfach, April 1988.}
 
\bibitem{BZ-combstrat}\ref
{A.~Bj\"orner \& G.~M.~Ziegler:}   
{Combinatorial stratification of complex arrangements,}
{{\sl J.\ Amer.\ Math.\ Soc.} {\bf 5} (1992), 105-149.}
 
\bibitem{BZ-reflect}\ref
{A.~Bj\"orner \& G.~M.~Ziegler:}   
{Reflections in oriented matroids and finite Coxeter groups,}
{in preparation.*}
 
\bibitem{Bland1}\ref
{R.~G.~Bland:}   
{Complementary orthogonal subspaces of $\R^n$ and orientability of matroids,} 
{Ph.D.~Thesis, Cornell University 1974, 80 pages.}
 
\bibitem{Bland2}\ref
{R.~G.~Bland:}  
{A combinatorial abstraction of linear programming,}
{{\sl J.~Combinatorial Theory} Ser.~B {\bf 23} (1977), 33-57.}
 
\bibitem{Bland3}\ref
{R.~G.~Bland:}   
{New finite pivoting rules for the simplex method,}
{{\sl Math.~Operations Research} {\bf 2} (1977), 103-107.}
 
\bibitem{Bland4}\ref
{R.~G.~Bland:}   
{Linear programming duality and Minty's lemma,} 
{preprint, Cornell University 1980, 43 pages.}
 
\bibitem{BlandCho}\ref
{R.~G.~Bland \& D.~E.~Cho,}   
{Balancing configurations in $\R^d$ by reflection of points,} 
{preprint, Cornell University 1987, 60 pages.}
 
\bibitem{BlandDietrich1}\ref
{R.~G.~Bland \& B.~L.~Dietrich:}   
{A unified interpretation of several combinatorial dualities,}
{preprint, Cornell University 1987, 41 pages.}
 
\bibitem{BlandDietrich2}\ref
{R.~G.~Bland \& B.~L.~Dietrich:}   
{An abstract duality,}
{{\sl Discrete Math.} {\bf 70} (1988), 203-208.}
 
\bibitem{BlandJensen}\ref
{R.~G.~Bland \& D.~L.~Jensen:}   
{Weakly oriented matroids,} 
{preprint, Cornell University 1987, 40 pages.}
 
\bibitem{BlandKoSturmfels}\ref
{R. G. Bland, C. W. Ko \& B. Sturmfels,}
{A nonextremal Camion basis,}
{{\sl Linear Alg.\ Applications} {\bf 187} (1993), 195-199.} 

\bibitem{BlandLasVergnas}\ref
{R.~G.~Bland \& M.~Las Vergnas:}  
{Orientability of matroids,} 
{{\sl J.\ Combinatorial Theory} Ser.~B {\bf 24} (1978), 94-123.}
 
\bibitem{BlandLasVergnas2}\ref
{R.~G.~Bland \& M.~Las Vergnas:}   
{Minty colorings and orientations of matroids,}
{{\sl Annals of the New York Acadademy of Sciences} {\bf 319} (1979), 86-92.}

\bibitem{BlassSagan}\ref
{A.~Blass \& B.~E.~Sagan:} 
{Bijective proofs of two broken circuit theorems,}
{{\sl J.\ Graph Theory} {\bf 10} (1986), 15-21.}
 
\bibitem{BlindBlind}\ref
{G.~Blind \& R.~Blind:} 
{Convex polytopes without triangular faces,} 
{{\it Israel J.\ Math.} {\bf 71} (1990), 129-134.}
 
\bibitem{BlindMani}\ref
{R.~Blind \& P.~Mani:}  
{On puzzles and polytope isomorphism,} 
{{\sl Aequationes Math.} {\bf 34} (1987), 287-297.}
 
\bibitem{Blumenthal}\ref
{L.~M.~Blumenthal:}  
{Theory and Applications of Distance Geometry,} 
{Oxford University Press (Clarendon Press) 1953; reprinted by Chelsea,
New York 1970.}

\bibitem{Bohne1}\ref
{J.~Bohne:}
{A characterization of oriented matroids in terms of conformal sequences,}
{{\sl Bayreuther Math.\ Schriften} {\bf 40} (1992), 1-5.}

\bibitem{Bohne2}\ref
{J.~Bohne:}
{Eine kombinatorische Analyse zonotopaler Raumaufteilungen,}
{Dissertation, Bielefeld 1992; 
Preprint 92-041, SFB~343, Universit\"at Bielefeld 1992, 100~pages.}

\bibitem{BohneDress}\ref
{J.~Bohne \& A.~W.~M.~Dress:} 
{Penrose tilings and oriented matroids,}
{in preparation.}*
 
\bibitem{BohneDressFischer}\ref
{J.~Bohne \& A.~W.~M.~Dress \& S.~Fischer:} 
{A simple proof for De Bruijn's dualization principle,}
{in: ``Proceedings of the Raj Chandra Bose Memorial Conference
on Combinatorial Mathematics and Applications'', Calcutta, 
India, {\sl Sankhya}, Ser.~A, {\bf 54}, 77-84.}
 
\bibitem{Bokowski1}\ref
{J.~Bokowski:}   
{A non-polytopal sphere,}
{preprint 1978, 4~pages.}

\bibitem{Bokowski2}\ref
{J.~Bokowski:}  
{Geometrische Realisierbarkeitsfragen -- 
Chirotope und orientierte Matroide,}
{in: Proc.\ 3.\ Kolloq.\ ``Diskrete Geometrie'',
Salzburg 1985, 53-57.}
 
\bibitem{Bokowski3}\ref
{J.~Bokowski:}  
{Aspects of computational synthetic geometry; II.~Combinatorial 
complexes and their geometric realization -- an algorithmic approach,} 
{in: {\sl Computer-aided geometric reasoning} (H.H.~Crapo, ed.),
INRIA Rocquencourt, France, June 1987.}
 
\bibitem{Bokowski4}\ref
{J.~Bokowski:}   
{A geometric realization without self-intersections
does exist for Dyck's regular map,}
{{\sl Discrete Comput.~Geometry} {\bf 4} (1989), 583-589.}
 
\bibitem{Bokowski5}\ref
{J.~Bokowski:} 
{On the generation of oriented matroids with prescribed topes,}
{preprint 1291, TH Darmstadt 1990, 11 pages.}
 
\bibitem{Bokowski6}\ref
{J.~Bokowski:}
{On the Las Vergnas conjecture concerning simplicial cells in
pseudo-plane arrangements,}
{preprint, 5~pages, 1991.}

\bibitem{Bokowski6a}\ref
{J.~Bokowski:}
{On the geometric flat embedding of abstract complexes with symmetries,}
{in: ``Symmetry of discrete mathematical structures and their symmetry groups,
a collection of essays'' (K.-H.~Hoffmann, R.~Wille, eds.)
Research and Exposition in Mathematics, Vol.~15, 
Heldermann Verlag, Berlin 1991, pp.~1-48.}

\bibitem{Bokowski-handbook}\ref
{J.~Bokowski:}
{Oriented matroids,}
{Chapter 2.5 in: {\sl Handbook of Convex Geometry} 
(eds.\ P.~Gruber, J.~Wills), North-Holland, Amsterdam, 1993, 555-602.}

\bibitem{Bokowski7}\ref
{J.~Bokowski:}
{On recent progress in computational synthetic geometry,}
{in: {\sl Polytopes: Abstract, Convex and Computational}
(T.~Bisztriczky, P.~McMullen, and A.~Weiss, eds.),
Proc.\ NATO Advanced Study Institute, Toronto 1993, 
Kluwer Academic Publishers 1994, pp.~335-358.}

\bibitem{Bokowski8}\ref
{J.~Bokowski:}
{On the construction of equifacetted $3$-spheres,}
{in: Proc.\ Conference ``Invariant Methods in Discrete and
Computational Geometry,'' Williamstadt, Cura\c cao 1994
(N.~White, ed.), Kluwer Academic Publishers, Dordrecht 1995, 301-312.}


\bibitem{Bokowski9}\ref
{J.~Bokowski:}
{Finite point sets and oriented matroids: Combinatorics in geometry,}
{in: ``Learning and Geometry: Computational Approaches''
(D.W.~Kuecker, C.H.~Smith, eds.), Progress in Computer Science, Vol.~14,
Birkh\"auser Boston 1996, pp.~67-96.}

\bibitem{BokowskiBrehm}\ref
{J.~Bokowski \& U.~Brehm:}   
{A new polyhedron of genus $3$ with $10$ vertices,} 
{in: Papers Int.\ Conf.\ {\sl Intuitive Geometry} 
(K.~B\"or\"oczky, G.~Fejes T\'oth, eds.), Si\'ofok/Hungary 1985,
{\sl Colloquia Math.\ Soc.\ J\'anos Bolyai} {\bf 48} (1985), 105-116.}
 
\bibitem{BokowskiEggert}\ref
{J.~Bokowski \& A.~Eggert:}   
{All realization of M\"obius' torus with $7$ vertices,}
{{\sl Structural Topology} {\bf 17} (1991), 59-78.}
 
\bibitem{BokowskiEwaldKleinschmidt}\ref
{J.~Bokowski, G.~Ewald \& P.~Kleinschmidt:}
{On combinatorial and affine automorphisms of polytopes,}
{{\sl Israel J.\ Math.} {\bf 47} (1984), 123-130.}

\bibitem{BokowskiGarms}\ref
{J.~Bokowski \& K.~Garms:} 
{Altshuler's sphere $M_{425}^{10}$ is not polytopal,}
{{\sl European J.~Combinatorics} {\bf 8} (1987), 227-229.}
 
\bibitem{BokowskiGdO1}\ref
{J.~Bokowski \& A.~Guedes de Oliveira:} 
{Simplicial convex $4$-polytopes do not have the isotopy property,}
{{\sl Portugaliae Mathematica} {\bf 47} (1990), 309-318.}
 
\bibitem{BokowskiGdO2}\ref
{J.~Bokowski \& A.~Guedes de Oliveira:} 
{Invariant theory-like theorems for matroids and oriented matroids,}
{{\sl Advances Math.} {\bf 109} (1994), 34-44.}

\bibitem{BokowskiGdORichter}\ref
{J.~Bokowski, A.~Guedes de Oliveira \& J.~Richter:} 
{Algebraic varieties characterizing matroids and oriented matroids,}
{Advances in Mathematics {\bf 87} (1991), 160-185.}
 
\bibitem{BokowskiGdOTV}\ref
{J.~Bokowski, A.~Guedes de Oliveira, U. Thiemann \& A. Veloso da Costa:}
{On the cube problem of Las Vergnas,}
{{\sl Geometriae Dedicata} {\bf 63} (1996), 25-43.}

\bibitem{BokowskiLaffailleRichterG}\ref
{J. Bokowski, G. Laffaille \& J. Richter-Gebert: } 
{Classification of non-stretchable pseudoline ar\-range\-ments 
and related properties,}
{in preparation, 1991.}*

\bibitem{BokowskiLaffailleRichter}\ref
{J.~Bokowski \& W.~Kollewe:} 
{On representing contexts in line arrangements,}
{{\sl Order} {\bf 8} (1992), 393-403.}
 
\bibitem{BokowskiRichter}\ref
{J.~Bokowski \& J.~Richter:} 
{On the finding of final polynomials,}
{{\sl European J.~Combinatorics} {\bf 11} (1990), 21-34.}
 
\bibitem{BokowskiRG1}\ref
{J.~Bokowski \& J.~Richter-Gebert:} 
{On the classification of non-realizable oriented matroids,
Part I: Generation,}
{preprint 1283, TH Darmstadt 1990, 17 pages.}
 
\bibitem{BokowskiRG2}\ref
{J.~Bokowski \& J.~Richter-Gebert:} 
{On the classification of non-realizable oriented matroids,
Part II: Properties,} 
{preprint, TH Darmstadt 1990, 22 pages.}
  
\bibitem{BokowskiRG3}\ref
{J.~Bokowski \& J.~Richter-Gebert:} 
{A new Sylvester-Gallai configuration representing
the 13-point projective plane in $\R^4$,}
{{\sl J.~Combinatorial Theory} Ser.~B, {\bf 54} (1992), 161-165.}
 
\bibitem{BokowskiRGSchindler}\ref
{J.~Bokowski, J.~Richter-Gebert \& W.~Schindler:} 
{On the distribution of order types,} 
{{\sl Computational Geometry: Theory and Applications} {\bf 1} (1992), 
127-142.}
 
\bibitem{BokowskiRichterSturmfels}\ref
{J.~Bokowski, J.~Richter \& B.~Sturmfels:}   
{Nonrealizability proofs in computational geometry,} 
{{\sl Discrete Comput.\ Geometry} {\bf 5} (1990), 333-350.}

\bibitem{BokowskiRoudneffStrempel}\ref
{J. Bokowski, J.-P.~Roudneff \& T.-K.~Strempel:}
{Cell decompositions of the projective plane with Petrie
polygons of constant length,}
{{\sl Discrete Comput.\ Geometry} {\bf 17} (1997), 377-392.}

\bibitem{BokowskiSchuchert1}\ref 
{J.~Bokowski \& P.~Schuchert:}
{Altshuler's sphere $M^9_{963}$ revisited,}
{{\sl SIAM J.\ Discrete Math.} {\bf 8} (1995), 670-677.}

\bibitem{BokowskiSchuchert2}\ref
{J.~Bokowski \& P.~Schuchert:}
{Equifacetted $3$-spheres as topes of nonpolytopal matroid polytopes,}
{in: ``The L\'aszl\'o Fejes T\'oth Festschrift'' 
(I.~B\'ar\'any, J~Pach, eds.),
{\sl Discrete Comput.\ Geometry} {\bf 13} (1995), 347-361.}

\bibitem{BokowskiShemer}\ref
{J.~Bokowski \& I.~Shemer:}   
{Neighborly $6$-polytopes with $10$ vertices,} 
{{\sl Israel J.\ Math.} {\bf 58} (1987), 103-124.}
 
\bibitem{BokowskiSturmfels1}\ref
{J.~Bokowski \& B.~Sturmfels:}   
{Problems of geometrical realizability --- oriented matroids and chirotopes,}
{preprint 901, TH Darmstadt 1985, 20 pages.}
 
\bibitem{BokowskiSturmfels2}\ref
{J.~Bokowski \& B.~Sturmfels:}   
{Programmsystem zur Realisierung orientierter Matroide,} 
{Programmdokumentation, {\sl Preprints in Optimization} 85.22, 
Universt\"at K\"oln 1985, 33 pages.}
 
\bibitem{BokowskiSturmfels3}\ref
{J.~Bokowski \& B.~Sturmfels:}   
{On the coordinatization of oriented matroids,}
{{\sl Discrete Comput.~Geometry} {\bf 1} (1986), 293-306.}
 
\bibitem{BokowskiSturmfels4}\ref
{J.~Bokowski \& B.~Sturmfels:}   
{Reell realisierbare orientierte Matroide, }
{{\sl Bayreuther Math.~Schriften} {\bf 21} (1986), 1-13.}
 
\bibitem{BokowskiSturmfels5}\ref
{J.~Bokowski \& B.~Sturmfels:}   
{Polytopal and nonpolytopal spheres. An algorithmic approach,} 
{{\sl Israel J.~Math.} {\bf 57} (1987), 257-271.}
 
\bibitem{BokowskiSturmfels6}\ref
{J.~Bokowski \& B.~Sturmfels:}   
{An infinite family of minor-minimal nonrealizable 3-chirotopes,} 
{{\sl Math.\ Zeitschrift} {\bf 200} (1989), 583-589.}
 
\bibitem{BokowskiSturmfels7}\ref
{J.~Bokowski \& B.~Sturmfels:}   
{Computational Synthetic Geometry,}
{Lecture Notes in Mathematics {\bf 1355} (1989), Springer, Heidelberg.}

\bibitem{BorovikGelfand}\ref
{A.~V.~Borovik \& I.~M.~Gelfand:}
{$WP$-matroids and thin Schubert cells on Tits systems,}
{{\sl Advances Math.} {\bf 103} (1994), 162-179.}
 
\bibitem{BorovikGelfandWhite}\ref
{A.~V.~Borovik, I.~M.~Gelfand \& N.~L.~White:}
{Boundaries of Coxeter matroids,}
{{\sl Advances in Math.} {\bf 120} (1996), 258-264.}

\bibitem{BorovikGelfandWhite2}\ref
{A.~V.~Borovik, I.~M.~Gelfand \& N.~L.~White:}
{On exchange properties for Coxeter matroids and oriented matroids,}
{{\sl Discrete Math.} {\bf 179} (1998), 59-72.}

\bibitem{Bourbaki}\ref
{N.~Bourbaki:}  
{Groupes et Alg\`ebres de Lie, Chap.~4,5, et 6,}
{Hermann, Paris 1968.}
 
\bibitem{Brown}\ref
{K.~S.~Brown:}  
{Buildings,} 
{Springer, New York 1989.}
 
\bibitem{BrownDiaconis}\ref
{K. S. Brown \& P. Diaconis:}
{Random walks and hyperplane arrangements,}
{Preprint, Cornell 1998, 39~pages.}

\bibitem{BruggesserMani}\ref
{H.~Bruggesser \& P.~Mani:} 
{Shellable decompositions of cells and spheres,}
{{\sl Math.~Scand.} {\bf 29} (1971), 197-205.}
 
\bibitem{BrylawskiZ}\ref
{T.~H.~Brylawski \& G.~M.~Ziegler:}  
{Topological representation of dual pairs of oriented matroids,} 
{Special issue on ``Oriented Matroids'' 
(eds.\ J.~Richter-Gebert, G.~M.~Ziegler), 
{\sl Discrete Comput.\ Geometry} (3) {\bf 10} (1993), 237-240.}

\bibitem{BuchiFenton}\ref
{J.~R.~Buchi \& W.~E.~Fenton:}  
{Large convex sets in oriented matroids,}
{{\sl J.~Combinatorial Theory} Ser.~B {\bf 45} (1988), 293-304.}
 
\bibitem{Buck}\ref
{R.~C.~Buck:} 
{Partitions of space,}
{{\sl Amer.\ Math.\ Monthly} {\bf 50} (1943), 541-544.}
 
\bibitem{Camion}\ref
{P.~Camion:}  
{Matrices totalement unimodulaires et probl\`emes combinatoires,} 
{Thesis, Univ.\ Brussels 1963.}
 
\bibitem{Camion2}\ref
{P.~Camion:}  
{Modules unimodulaires,} 
{{\sl J.~Combinatorial Theory} {\bf 4} (1968), 301-362.} 
 
\bibitem{Canham}\ref
{R. J.~Canham:}
{\ }  
{Ph.D.~Thesis, Univ.~of East Anglia, Norwich 1971.}
 
\bibitem{Canny}\ref
{J.~Canny:}  
{Some algebraic and geometric computations in PSPACE,} 
{{\sl Proc.\ 20th ACM Symposium on Theory of Computing}, Chicago, May 1988.}
 
\bibitem{CGPPSW}\ref
{S.~E.~Cappell, J.~E.~Goodman, J.~Pach, R.~Pollack, M.~Sharir \& R.~Wenger:} 
{The combinatorial complexity of hyperplane transversals,}
{{\sl Proc.\ 6th ACM Annual Symposium\ on Computational Geo\-me\-try}, 
(Berkeley, June 1990), ACM 1990, pp.~83-91.}
     
\bibitem{CGPPSW2}\ref
{S. E. Cappell, J. E. Goodman, J. Pach, R. Pollack, M. Sharir \& R. Wenger:} 
{Com\-mon tan\-gents and common transversals,}
{{\sl Advances Math.} {\bf 106} (1994), 198-215.}

\bibitem{Cara}\ref
{C.~Carath\'eodory:}  
{\"Uber den Variabilit\"atsbereich der
Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen,}
{{\sl Math.\ Annalen} {\bf 64} (1904), 95-115.}

\bibitem{Chaiken1}\ref
{S. Chaiken:}
{Oriented matroid pairs, theory and an electric application,}
{in: ``Matroid Theory'' (J. E. Bonin, J. G. Oxley, B. Servatius, eds.),
{\sl Contemporary Math.} {\bf 197}, Amer.\ Math.\ Soc. 1995, 313-331.}
 
\bibitem{Chaiken2}\ref
{G.~D.~Chakerian:}  
{Sylvester's problem on collinear points and a relative,}
{{\sl American Math.\ Monthly} {\bf 77} (1970), 164-167.}
 
\bibitem{Cheung}\ref
{A.~L.~C.~Cheung:}  
{Adjoints of a geometry, }
{{\sl Canadian Math.~Bulletin,} {\bf 17} (1974), 363-365.}
 
\bibitem{CheungCrapo}\ref
{A.~L.~C.~Cheung \& H.~H.~Crapo:}  
{A combinatorial perspective on algebraic geometry,} 
{{\sl Advances in Math.} {\bf 20} (1976), 388-414.}
 
\bibitem{Chvatal}\ref
{V.~Chv\'atal:}  
{Linear Programming,}
{Freeman, New York 1983.}

\bibitem{Clausen}\ref
{J.~Clausen:}  
{A note on the Edmonds-Fukuda pivoting rule for simplex algorithms,} 
{{\sl European J.~Operational Research} {\bf 29} (1987), 378-383.}
 
\bibitem{ClausenTerlaky}\ref
{J.~Clausen \& T.~Terlaky:} 
{On the feasibility of the Edmonds-Fukuda pivoting rule for 
oriented matroid programming,} 
{preprint 1987, 16 pages.}
 
\bibitem{Collins}\ref
{C.~Collins:}  
{Quantifier elimination for real closed fields
by cylindrical algebraic decomposition,} 
{in: {\sl Automata Theory and Formal Languages} (H.~Brakhage, ed.), 
Springer Lecture Notes in Computer Science {\bf 33} (1975), 134-163.}
 
\bibitem{CookeFinney}\ref
{G.~E.~Cooke \& R.~L.~Finney:} 
{Homology of Cell Complexes,}
{Princeton University Press 1967.}

\bibitem{Cordovil1}\ref
{R.~Cordovil:}   
{Sur les orientations acycliques des g\'eom\'etries orient\'ees de rang $3$,}
{Actes du {\sl Colloque International sur la Theorie des Graphes 
et la Combinatoire}
(Marseilles 1981), {\sl Ann.~Discrete Math.} {\bf 9} (1980), 243-246.}
 
\bibitem{Cordovil2}\ref
{R.~Cordovil:}   
{Sur l'\'evaluation $t(M;2,0)$ du polyn\^ome de Tutte d'un matro\"\i de et 
une conjecture de B.~Gr\"unbaum relative aux arrangements de droites du plan,}
{{\sl European J.~Combinatorics} {\bf 1} (1980), 317-322.}
 
\bibitem{Cordovil3}\ref
{R.~Cordovil:}   
{Quelques propri\'et\'es alg\'ebriques des matro\"\i des,} 
{Th\`ese, Universit\'e Paris VI, 1981, 167 pages.}
 
\bibitem{Cordovil4}\ref
{R.~Cordovil:}   
{Sur les matro\"\i des orient\'es de rang $3$ et les arrangements de
pseudodroites dans le plan projectif r\'eel,} 
{{\sl European J.~Combinatorics} {\bf 3} (1982), 307-318.}

\bibitem{Cordovil5}\ref
{R.~Cordovil:}  
{Sur un th\'eor\`eme de s\'eparation
des matro\"\i des orient\'es de rang $3$,}
{{\sl Discrete Math.} {\bf 40} (1982), 163-169.}
 
\bibitem{Cordovil6}\ref
{R.~Cordovil:}  
{The directions determined by $n$ points in the plane: a matroidal 
generalization,} 
{{\sl Discrete Math.} {\bf 43} (1983), 131-137.}
 
\bibitem{Cordovil7}\ref
{R.~Cordovil:}  
{Oriented matroids of rank three and arrangements of pseudolines,}
{{\sl Ann.~Discrete Math.} {\bf 17} (1983), 219-223.}
 
\bibitem{Cordovil8}\ref
{R.~Cordovil:}  
{Oriented matroids and geometric sorting,} 
{{\sl Canad.~Math.~Bull.} {\bf 26} (1983), 351-354.}
 
\bibitem{Cordovil9}\ref
{R.~Cordovil:} 
{A combinatorial perspective on non-Radon partitions,} 
{{\sl J.~Combinatorial Theory}, Ser.~A {\bf 38} (1985), 38-47.
(Erratum, ibid.~{\bf 40}, 194.)}
 
\bibitem{Cordovil10}\ref
{R.~Cordovil:} 
{On the number of lines determined by $n$ points,} 
{preprint 1986, 16 pages.}
 
\bibitem{Cordovil11}\ref
{R.~Cordovil:}  
{Polarity and point extensions in oriented matroids,} 
{{\sl Linear Algebra and its Appl.} {\bf 90} (1987), 15-31.}
   
\bibitem{Cordovil12}\ref
{R.~Cordovil:}
{On the homotopy type of the Salvetti complexes 
determined by simplicial arrangements,}
{{\sl European J.\ Combinatorics} {\bf 15} (1994), 207-215.}

\bibitem{Cordovil13}\ref
{R.~Cordovil:}
{On the center of the fundamental group of the complement of an
arrangement of hyperplanes,}
{{\sl Portugaliae Math.} {\bf 51} (1994), 363-373.}

\bibitem{CordovilDaSilva1}\ref
{R.~Cordovil \& I.~P.~da Silva:}   
{A problem of McMullen on the projective equivalence of polytopes,} 
{{\sl European J.~Combinatorics} {\bf 6} (1985), 157-161.}
 
\bibitem{CordovilDaSilva2}\ref
{R.~Cordovil \& I.P.~da Silva:} 
{Determining a matroid polytope by non-Radon partitions,} 
{{\sl Linear Algebra Appl.} {\bf 94} (1987), 55-60.}
 
\bibitem{CordovilDuchet1}\ref
{R.~Cordovil \& P.~Duchet:}   
{S\'eparation par une droite dans les matro\"\i des orient\'es de rang~$3$,} 
{{\sl Discrete Math.} {\bf 62} (1986), 103-104.}
 
\bibitem{CordovilDuchet2}\ref
{R.~Cordovil \& P.~Duchet:}   
{Cyclic polytopes and oriented matroids,}
{preprint 1987, 17 pages.}
 
\bibitem{CordovilDuchet3}\ref
{R.~Cordovil \& P.~Duchet:}   
{On the sign-invariance graphs of uniform oriented matroids,} 
{{\sl Discrete Math.} {\bf 79} (1989/90), 251-257.}

\bibitem{CordovilFachada}\ref
{R.~Cordovil \& J.~L.~Fachada:}
{Braid monodromy groups and wiring diagrams,}
{{\sl Bolletino U. M. I.} {\bf 9-B} (1995), 399-416.}

\bibitem{CordovilFukuda}\ref
{R.~Cordovil \& K.~Fukuda:}   
{Oriented matroids and combinatorial manifolds,} 
{{\sl European J.\ Combinatorics} {\bf 14} (1993), 9-15.}

\bibitem{CordovilFukudaGdO}\ref
{R.~Cordovil, K.~Fukuda \& A.~Guedes de Oliveira:} 
{On the cocircuit graph of an oriented matroid,}
{preprint 1991, 11 pages.}

\bibitem{CordovilGdO}\ref
{R.~Cordovil \& A.~Guedes de Oliveira:}
{A note on the fundamental group of the Salvetti complex determined by an 
oriented matroid,}
{{\sl Europ.\ J.~Combinatorics} {\bf 13} (1992), 429-437.}

\bibitem{CordovilGdOLV}\ref
{R. Cordovil, A. Guedes de Oliveira \& M. Las Vergnas:}
{A generalized Desargues configuration and the pure braid group,}
{{\sl Discrete Math.} {\bf 160} (1996), 105-113.}

\bibitem{CordovilGdOMoreira}\ref
{R.~Cordovil, A.~Guedes de Oliveira \& M.~L.~Moreira:}  
{Parallel projection of matroid spheres,}
{{\sl Portugaliae Mathematica} {\bf 45} (1988), 337-346.}
 
\bibitem{CordovilMandel}\ref
{R.~Cordovil, M.~Las Vergnas \& A.~Mandel:}   
{Euler's relation, M\"obius functions and matroid identities,} 
{{\sl Geometriae Dedicata} {\bf 12} (1982), 147-162.}
 
\bibitem{CordovilMoreira}\ref
{R.~Cordovil \& M.~L.~Moreira:} 
{A homotopy theorem on oriented matroids,} 
{{\sl Discrete Math.} {\bf 111} (1993), 131-136.}

\bibitem{CHS1}\ref
{M.-F.~Coste-Roy, J.~Heintz \& P.~Solerno:} 
{On the complexity of semialgebraic sets,} 
{preprint 1989.}

\bibitem{CHS2}\ref
{M.-F.~Coste-Roy, J.~Heintz \& P.~Solerno:}
{Description of the connected components of a semialgebraic set in
single exponential time,}
{{\sl Discrete Comput.\ Geometry} {\bf 11} (1994), 121-140.}
 
\bibitem{Coxeter1}\ref
{H.~S.~M.~Coxeter:}  
{The complete enumeration of finite groups of the 
form $R_i^2=(R_iR_j)^{k_{ij}}=1$,} 
{{\sl J.~London Math.~Soc.} {\bf 10} (1935), 21-25.}
 
\bibitem{Coxeter2}\ref
{H.~S.~M.~Coxeter:}  
{The classification of zonohedra by means of projective diagrams,} 
{{\sl Journal de Math.\ Pures Appl.} {\bf 41} (1962), 137-156.}
 
\bibitem{Coxeter3}\ref
{H.~S.~M.~Coxeter:}  
{Regular Polytopes (third edition),} 
{Dover, New  York 1973.}
 
\bibitem{Crapo}\ref
{H.~H.~Crapo:}  
{Single-element extensions of matroids,}
{{\sl J.\ Research Nat.\ Bureau Standards} {\bf 69B} (1965), 55-65.}
 
\bibitem{CrapoLaumond}\ref
{H.~H.~Crapo \& J.-P.~Laumond:}  
{Hamiltonian cycles in Delaunay complexes,} 
{in: {\sl Geometry and Robotics} (J.-D.~Boinsonnat et J.-P.~Laumond, eds.),
Lecture Notes in Computer Science, Springer 1989, 292-305.}
 
\bibitem{CrapoPenne}\ref
{H.~H.~Crapo \& R.~Penne:}
{Chirality and the isotopy classification of skew lines in 
projective $3$-space,}
{{\sl Advances in Math.} {\bf 103} (1994), 1-106.}
  
\bibitem{CrapoRG1}\ref
{H. Crapo \& J. Richter-Gebert:}
{Automatic proving of geometric theorems,}
{in: Proc.\ Conference ``Invariant Methods in Discrete and
Computational Geometry,'' Williamstadt, Curacao 1994 (N.~White, ed.),
Kluwer Academic Publishers, Dordrecht 1995, 167-196.}

\bibitem{CrapoRG2}\ref
{H. Crapo \& J.~Richter-Gebert:}
{CINDERELLA Computer Interactive Drawing Environment,} %% Really ELLegant.
{work in progress, 1992/94.}

\bibitem{CrapoRota}\ref
{H.~H.~Crapo \& G.-C.~Rota:}  
{Combinatorial Geometries,} 
{MIT Press 1970.}

\bibitem{CrapoSenechal}\ref
{H.~Crapo \& M. Senechal:}
{Tilings by related zonotopes,}
{preprint 1995, 14 pages;
{\sl Mathematical and Computer Modelling}, to appear.}

\bibitem{CrippenHavel}\ref
{G.~M.~Crippen \& T.~F.~Havel:} 
{Distance Geometry and Molecular Conformation,}
{Research Studies Press, Taunton/England, 1988.}

\bibitem{CsimaSawyer}\ref
{J.~Csima \& E.T. Sawyer:} 
{There exist $6n/13$ ordinary points,}
{{\sl Discrete Comput.\ Geometry} {\bf 9} (1993), 187-202.}

\bibitem{DanarajKlee1}\ref
{G.~Danaraj \& V.~Klee:} 
{Shellings of spheres and polytopes,}
{{\sl Duke Math.\ J.} {\bf 41} (1974), 443-451.}
 
\bibitem{Dantzig}\ref
{G.~B.~Dantzig:}  
{Linear Programming and Extensions,} 
{Princeton University Press 1963.}
 
\bibitem{daSilva1}\ref
{I.~P.~da Silva:} 
{Quelques propri\'et\'es des matro\"\i des orient\'es,} 
{Dissertation, Universit\'e Paris VI, 1987, 131 pages.}
 
\bibitem{daSilva2}\ref
{I.~P.~da Silva:} 
{An axiomatic for the set of maximal vectors of an oriented matroid
based on symmetry properties of the set of vectors,} 
{preprint 1988, 18~pages.}
 
\bibitem{daSilva3}\ref
{I.~P.~F.~da Silva:} 
{Axioms for maximal vectors of an oriented matroid:
a combinatorial characterization of the regions determined by
an arrangement of pseudohyperplanes,}
{{\sl European J.~Combinatorics} {\bf 16} (1995), 125-145.}

\bibitem{daSilva4}\ref
{I.~P.~F.~da Silva:}
{On fillings of $2N$-gons with rhombi,}
{{\sl Discrete Math.} {\bf 111} (1993), 137-144.} 

\bibitem{daSilva5}\ref
{I.~P.~F.~da Silva:} 
{On inseparability graphs of matroids having exactly 
one class of orientations,}
{preprint 1993, 7~pages.}

\bibitem{daSilva6}\ref
{I.~P.~F.~da Silva:} 
{An intersection property defining series-parallel networks,}
{preprint 1993, 18~pages; preprint 1994, 30~pages.}

\bibitem{Daverman}\ref
{R.~J.~Daverman:} 
{Decompositions of Manifolds,}
{Academic Press, 1986.}
 
\bibitem{Deligne}\ref
{P.~Deligne:} 
{Les immeubles des groupes de tresses g\'en\'eralis\'es,}
{{\sl Inventiones Math.} {\bf 17} (1972), 273-302.}

\bibitem{DeLoera}\ref
{J. A. De Loera:}
{Nonregular triangulations of products of simplices,}
{{\sl Discrete Comput.\ Geometry} {\bf 15} (1996), 253-264.}

\bibitem{DeLHSS}\ref
{J. A. De Loera, S. Ho\c{s}ten, F. Santos \& B. Sturmfels,}
{The polytope of all triangulations of a point configuration,}
{{\sl Documenta Mathematica} {\bf 1} (1996), 103-119.}

\bibitem{DeLoeraMorris}\ref
{J. A. De Loera:}
{$Q$-matrix recognition via secondary and universal polytopes,}
{Preprint 1997, 24 pages.}

\bibitem{DeLoeraSantosUrrutia}\ref
{J. A. De Loera, F. Santos \& J. Urrutia:}
{The number of geometric bistellar neighbors of a triangulation,}
{Preprint, 16~pages; {\sl Discrete Comput.\ Geometry}, to appear.}

\bibitem{DelsarteKamp}\ref
{P.~Delsarte \& Y.~Kamp:} 
{Low rank matrices with a given sign pattern,}
{{\sl SIAM J.\ Discrete Math.} {\bf 2} (1989), 51-63.}
 
\bibitem{DezaFukuda}\ref
{M.~Deza \& K.~Fukuda:}   
{On bouquets of matroids and orientations,} 
{{\sl Publ.\ R.I.M.S.} Kyoto University, Kokyuroku {\bf 587} (1986), 110-129.}
  
\bibitem{Dillencourt}\ref
{M.~B.~Dillencourt:} 
{An upper bound on the shortness exponent of inscribable polytopes,}
{{\sl J.~Combinatorial Theory}, Ser.~B {\bf 46} (1989), 66-83.}

\bibitem{DDH}\ref
{A.~Dreiding, A.~W.~M.~Dress \& H.~Haegi:}   
{Classification of mobile molecules by category theory,} 
{{\sl Studies in Physical and Theoretical Chemistry} {\bf 23} (1982), 39-58.}

\bibitem{DreidingWirth}\ref
{A.~Dreiding \& K.~Wirth:}  
{The multiplex. A classification of finite ordered point sets in oriented 
$d$-dimensional space, }
{{\sl Math.~Chemistry} {\bf 8} (1980), 341-352.}
 
\bibitem{Dress1}\ref
{A.~W.~M.~Dress:}   
{Duality theory for finite and infinite matroids with coefficients,}
{{\sl Advances in Math.} {\bf 59} (1986), 97-123.}
 
\bibitem{Dress2}\ref
{A.~W.~M.~Dress:}   
{Chirotops and oriented matroids,}
{{\sl Bayreuther Math.\ Schriften} {\bf 21} (1986), 14-68.}
 
\bibitem{DressScharlau}\ref
{A.~W.~M.~Dress \& R.~Scharlau:} 
{Gated sets in metric spaces,}
{{\sl Aequationes Mathematicae} {\bf 34} (1987), 112-200.}
 
\bibitem{DressWenzel1}\ref
{A.~W.~M.~Dress \& W.~Wenzel:} 
{Endliche Matroide mit Koeffizienten,}
{{\sl Bayreuther Math.\ Schrif\-ten} {\bf 26} (1988), 37-98.}
  
\bibitem{DressWenzel2}\ref
{A.~W.~M.~Dress \& W.~Wenzel:}  
{Geometric algebra for combinatorial geometries,}
{{\sl Advances in Math.} {\bf 77} (1989), 1-36.}
 
\bibitem{DressWenzel3}\ref
{A.~W.~M.~Dress \& W.~Wenzel:} 
{On combinatorial and projective geometry,} 
{{\sl Geometriae Dedicata} {\bf 34} (1990), 161-197.}
  
\bibitem{DressWenzel4}\ref
{A.~W.~M.~Dress \& W.~Wenzel:} 
{Grassmann-Pl\"ucker relations and matroids with coefficients,}
{{\sl Advances in Mathematics} {\bf 86} (1991), 68-110.}

\bibitem{DressWenzel5}\ref
{A.~W.~M.~Dress \& W.~Wenzel:} 
{Perfect matroids,}
{{\sl Advances in Math.} {\bf 91} (1992), 158-208.}
  
\bibitem{DressWenzel6}\ref
{A.~W.~M.~Dress \& W.~Wenzel:} 
{Valuated matroids,}
{{\sl Advances in Math.} {\bf 93} (1992), 214-250.}

\bibitem{Duchet}\ref
{P. Duchet:}
{Convexity in combinatorial structures,}
{in: ``Abstract analysis,'' Proc.\ 14th Winter School, Srni/Czech. 1986, 
{\sl Suppl.\ Rend.\ Circ.\ Mat.\ Palermo} II.~Ser.\ {\bf 14} (1987), 261-293.}

\bibitem{Edelman1}\ref
{P.~H.~Edelman:}   
{Meet-distributive lattices and the anti-exchange closure,} 
{{\sl Algebra Universalis} {\bf 10} (1980), 290-299.}

\bibitem{Edelman2}\ref
{P.~H.~Edelman:}    
{The lattice of convex sets of an oriented matroid,} 
{{\sl J.~Combinatorial Theory} Ser.~B {\bf 33} (1982), 239-244.}
 
\bibitem{Edelman3}\ref
{P.~H.~Edelman:}    
{The acyclic sets of an oriented matroid,} 
{{\sl J.~Combinatorial Theory} Ser.~B {\bf 36} (1984), 26-31.}
 
\bibitem{Edelman4}\ref
{P.~H.~Edelman:}    
{A partial order on the regions of $\R^n$ dissected by hyperplanes,}
{{\sl Transactions Amer.\ Math.\ Soc.} {\bf 283} (1984), 617-631.}
  
\bibitem{EdelmanGreene}\ref
{P.~H.~Edelman \& C.~Greene:}  
{Balanced tableaux,}
{{\sl Advances in Math.} {\bf 63} (1987), 42-99.}
  
\bibitem{EdelmanJamison}\ref
{P.~H.~Edelman \& R.~E.~Jamison:} 
{The theory of convex geometries,}
{{\sl Geometriae Dedicata} {\bf 19} (1985), 247-270.}

\bibitem{EdelmanRambauReiner}\ref
{P.~H.~Edelman, J. Rambau \& V.~Reiner:}
{On subdivision posets of cyclic polytopes,}
{MSRI Preprint 1997-030.}

\bibitem{EdelmanReiner1}\ref
{P.~H.~Edelman \& V.~Reiner:}
{Free arrangements and rhombic tilings,}
{{\sl Discrete Comput.\ Geometry} {\bf 15} (1996), 307-340;
erratum {\bf 17} (1997), 359.}

\bibitem{EdelmanReiner2}\ref
{P.~H.~Edelman \& V.~Reiner:}
{The higher Stasheff-Tamari posets,}
{{\sl Mathematika} {\bf 43} (1996), 127-154.}

\bibitem{EdelmanWalker}\ref
{P.~H.~Edelman \& J.W.~Walker:}   
{The homotopy type of hyperplane posets,}
{{\sl Proceedings Amer.\ Math.\ Soc.} {\bf 94} (1985), 329-332.}
 
\bibitem{Edelsbrunner}\ref
{H.~Edelsbrunner:}  
{Algorithms in Computational Geometry,} 
{Springer 1987.}
 
\bibitem{EdelsbrunnerGuibas}\ref
{H.~Edelsbrunner \& L.~J.~Guibas:}   
{Topologically sweeping an arrangement,} 
{{\sl J.~Computer and System Sciences} {\bf 38} (1989), 165-194.}

\bibitem{EdelsbrunnerMucke}\ref
{H.~Edelsbrunner \& E.~P.~M\"ucke:} 
{Simulation of simplicity: A technique to cope with
degenerate cases in geometric algorithms,}
{{\sl Fourth Annual ACM Symposium on Computational Geometry} 1988,
pp.~118-133.}

\bibitem{EdelsbrunnerORSeidel}\ref
{H.~Edelsbrunner, J.~O'Rourke \& R.~Seidel:}   
{Constructing arrangements of lines and hyperplanes with applications,} 
{{\sl SIAM J.\ Computing} {\bf 15} (1986), 341-363.}
 
\bibitem{EdelsbrunnerSeidel}\ref
{H.~Edelsbrunner \& R.~Seidel:}   
{Voronoi diagrams and arrangements,}
{{\sl Discrete Comput.~Geometry} {\bf 1} (1986), 25-44.}

\bibitem{EdelsbrunnerSeidelSharir}\ref
{H.~Edelsbrunner, R.~Seidel \& M.~Sharir:} 
{On the zone theorem for hyperplane arrangements,}
{{\sl SIAM J.\ Computing} {\bf 22} (1993), 418-429.}

\bibitem{EdelsbrunnerWelzl}\ref
{H.~Edelsbrunner \& E.~Welzl:} 
{On the number of line separations of a finite set in the plane,}
{{\sl J.~Combinatorial Theory} Ser.~A {\bf 38} (1985), 15-29.}
 
\bibitem{Edmonds}\ref
{J.~Edmonds:}  
{Submodular functions, matroids and certain polyhedra,}
{in: {\sl Combinatorial Structures and their Applications},
(H.\ Hanani, N.\ Sauer and J.\ Sch\"onheim, eds.)
Gordon and Breach, New York 1970, pp.~69-87.}

\bibitem{EdmondsFukuda}\ref
{J.~Edmonds \& K.~Fukuda:}   
{Oriented matroid programming,}
{Ph.D.~Thesis of K.\ Fukuda, University of Waterloo 1982, 223 pages.}
  
\bibitem{EdmondsLovaszMandel}\ref
{J.~Edmonds, L.~Lov\'asz \& A.~Mandel:}   
{Solution,} 
{{\sl Math.~Intelligencer} {\bf 2} (1980), 107.}

\bibitem{EdmondsMandel}\ref
{J.~Edmonds \& A.~Mandel:}  
{Topology of oriented matroids,}
{Abstract 758-05-9, {\sl Notices Amer.\ Math.\ Soc.} {\bf 25} (1978), A-510.}
 
\bibitem{EdmondsMandel2}\ref
{J.~Edmonds \& A.~Mandel:}   
{Topology of oriented matroids,}  
{Ph.D.~Thesis of A.~Mandel, University of Waterloo 1982, 333 pages.}

\bibitem{Elnitzky}\ref
{S. Elnitzky:}
{Rhombic tilings of polygons and classes of reduced words in Coxeter groups,}
{Ph.~D.\ Thesis, University of Michigan, 1993.}
  
\bibitem{Elnitzky2}\ref
{S. Elnitzky:}
{Rhombic tilings of polygons and classes 
of reduced words in Coxeter groups,}
{{\sl J. Combinatorial Theory} Ser.~A {\bf 77} (1997), 193-221.}

\bibitem{ErdosPurdy}\ref
{P.~Erd\H{o}s \& G.~Purdy:}   
{Some extremal problems in combinatorial geometry,} 
{in: {\sl Handbook of Combinatorics} (R.~Graham, M.~Gr\"otschel, 
L.~Lov\'asz, eds.), North-Holland/Elsevier, Amsterdam 1995, 809-874.}
 
\bibitem{ErdosSzekeres}\ref
{P.~Erd\H{o}s \& G.~Szekeres:} 
{A combinatorial problem in geometry,}
{{\sl Compositio Math.} {\bf 2} (1935), 463-470.}
 
\bibitem{EwaldKleinschmidtPS}\ref 
{G. Ewald, P. Kleinschmidt, U. Pachner \& C. Schulz:} 
{Neuere Entwicklungen in der kom\-bi\-na\-to\-ri\-schen Konvexgeometrie,}
{in: {\sl Contributions to Geometry} (J. T\"olke, J. Wills, eds.),
{Proc.\ Geo\-me\-try Sym\-po\-sium Siegen 1978},
Birkh\"auser 1979, pp.~131-163.}

\bibitem{Faigle}\ref
{U.~Faigle:}   
{Orthogonal sets, matroids, and theorems of the alternative,} 
{{\sl Bolletino Unione Mat.\ Ital.} VI.~Ser.\ {\bf 4-B} (1985), 139-153.}
 
\bibitem{Falk1}\ref
{M.~J.~Falk:} 
{Geometry and topology of hyperplane arrangements,}
{Ph.D.~Thesis, University of Wisconsin, Madison 1983, 103 pages.}
 
\bibitem{Falk2}\ref
{M.~J.~Falk:} 
{On the algebra associated with a geometric lattice,}
{{\sl Advances in Math.} {\bf 80} (1990), 152-163.}

\bibitem{Falk3}\ref
{M.~J.~Falk:}
{Homotopy types of line arrangements,}
{{\sl Inventiones Math.} {\bf 111} (1993), 139-150.}

\bibitem{Felsner}\ref
{S.~Felsner:}
{On the number of arrangements of pseudolines,}
{{\sl Discrete Comput.\ Geometry} {\bf 18} (1997), 257-267.}

\bibitem{FelsnerKriegel}\ref
{S. Felsner \& K. Kriegel:}
{Triangles in Euclidean arrangements,}
{Preprint, FU Berlin 1998, 10~pages.}

\bibitem{FelsnerWeil}\ref
{S. Felsner \& H. Weil:}
{Sweeps, arrangements and signotopes,}
{Preprint, FU Berlin 1998, 27~pages.}

\bibitem{Fenchel}\ref
{W.~Fenchel:}  
{Convexity through the ages,} 
{in: {\sl Convexity and its applications}
(P.~Gruber, J.~Wills, eds.), Birkh\"auser 1983, pp.~120-130.}
  
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{W.~E.~Fenton:}   
{Axiomatic of convexity theory,} 
{Ph.D.~Thesis, Purdue Univ.~1982, 98 pages.}
 
\bibitem{Fenton2}\ref
{W.~E.~Fenton:}   
{Completeness in oriented matroids,} 
{{\sl Discrete Math.} {\bf 66} (1987), 79-89.}
 
\bibitem{Finashin}\ref
{S.~M.~Finashin:}  
{Configurations of seven points in $\RP^3$,} 
{in: {\sl Topology and Geometry --- Rohlin Seminar} (O.Ya.~Viro, ed.),
Lecture Notes in Mathematics {\bf 1346} (1988), Springer, pp.~501-526.}
 
\bibitem{FolkmanLawrence}\ref
{J.~Folkman \& J.~Lawrence:}   
{Oriented matroids,} 
{{\sl J.~Combinatorial Theory}, Ser.~B {\bf 25} (1978), 199-236.}

\bibitem{ForgeRAlfonsin}\ref
{D. Forge \& J. L. Ram\'{\i}rez Alfons\'{\i}n:}
{Straight line arrangements in the real projective plane,}
{{\sl Discrete Comput.\ Geometry} {\bf 20} (1998), 155-161.}

\bibitem{ForgeSchuchert}\ref
{D. Forge \& P. Schuchert:}
{A set of points not projectively invariant to the vertices of a polytope,}
{Preprint 1996, 6~pages.}

\bibitem{Fukuda}\ref
{K.~Fukuda:}   
{Oriented matroids and linear programming,}
{(in Japanese), Proceedings of the 15th Symposium 
of the Operations Research Society of Japan, 1986, pp.~8-14.}
 
\bibitem{FukudaHanda1}\ref
{K.~Fukuda \& K.~Handa:}   
{Perturbations of oriented matroids and acycloids,} 
{preprint 1985, 18 pages.}
 
\bibitem{FukudaHanda2}\ref
{K.~Fukuda \& K.~Handa:}   
{Antipodal graphs and oriented matroids,}
{{\sl Discrete Math.} {\bf 111} (1993), 245-256.}
   
\bibitem{FukudaMatsui1}\ref
{K.~Fukuda \& T.~Matsui:}  
{On the finiteness of the crisscross method,} 
{{\sl European Journal of Operational Research} {\bf 52} (1991), 119-124.}

\bibitem{FukudaMatsui2}\ref
{K.~Fukuda \& T.~Matsui:} 
{Elementary inductive proofs for linear programming,}
{{\sl Publ.\ R.I.M.S.} Kyoto University, Kokyuroku 1989.}

\bibitem{FukudaST}\ref
{K.~Fukuda, S.~Saito \& A.~Tamura:} 
{Combinatorial face enumeration in arrangements and oriented matroids,} 
{{\sl Discrete Appl.\ Math.}.~{\bf 31} (1991), 141-149.}
 
\bibitem{FukudaSTT}\ref
{K.~Fukuda, S.~Saito, A.~Tamura \& T.~Tokuyama:} 
{Bounding the number of $k$-faces in arrangements of hyperplanes,} 
{{\sl Discrete Appl.\ Math.} {\bf 31} (1991), 151-165.}
 
\bibitem{FukudaTamura1}\ref
{K.~Fukuda \& A.~Tamura:}   
{Local deformation and orientation transformation in oriented matroids,} 
{{\sl Ars Combinatoria} {\bf 25A} (1988), 243-258.}
 
\bibitem{FukudaTamura2}\ref
{K.~Fukuda \& A.~Tamura:}   
{Local deformation and orientation transformation in oriented matroids II,} 
{preprint, Research Reports on Information Sciences B-212, 
Tokyo Institute of Technology 1988, 22 pages.}
 
\bibitem{FukudaTamura3}\ref
{K.~Fukuda \& A.~Tamura:}   
{Characterizations of $*$-families,} 
{{\sl J.~Combinatorial Theory}, Ser.~B {\bf 47} (1989), 107-110.}

\bibitem{FukudaTamura4}\ref
{K.~Fukuda \& A.~Tamura:} 
{Dualities in signed vector spaces,}
{{\sl Portugaliae Mathematica} {\bf 47} (1990), 151-165.}

\bibitem{FukudaTamuraT}\ref
{K.~Fukuda, A.~Tamura \& T.~Tokuyama:} 
{A theorem on the average number of subfaces 
in arrangements and oriented matroids,}
{{\sl Geom.\ Dedicata} {\bf 47} (1993), 129-142.}
 
\bibitem{FukudaTerlaky}\ref
{K.~Fukuda \& T.~Terlaky:}  
{A general algorithmic framework for quadratic programming and a generalization 
of Edmonds-Fukuda rule as a finite version of Van de Panne-Whinston method,} 
{preprint 1989, 18 pages.}

\bibitem{FukudaTerlaky2}\ref
{K.~Fukuda \& T.~Terlaky:}  
{Linear complementarity and oriented matroids,}
{{\sl J.~Operations Research Society of Japan} {\bf 35} (1992), 45-61.}

\bibitem{Fulkerson}\ref
{D.~R.~Fulkerson:}  
{Networks, frames, blocking systems,}
{in: Mathematics of the Decision Sciences, Part I,
(G.B.~Dantzig, A.F.~Vienot eds.) Lectures in Applied Mathematics {\bf 2}, 
Amer.\ Math.\ Soc.\ 1968, pp.~303-334.}

\bibitem{Gartner}\ref
{B.~G\"artner:}
{Set systems of bounded Vapnik-Chervonenkis dimension
and a relation to arrangements,}
{Diplomarbeit, FU Berlin 1991.}

\bibitem{GartnerWelzl}\ref
{B.~G\"artner \& E.~Welzl:}
{Vapnik-Chervonenkis dimension and (pseudo-)hyperplane arrangements,}
{{\sl Discrete Comput.\ Geometry} {\bf 12} (1994), 399-432.}

\bibitem{GHJ}\ref
{H.~G\"unzel, R.~Hirabayashi \& H.~T.~Jongen:}
{Multiparametric optimization: on stable singularities
occurring in combinatorial partition codes,}
{{\sl Control \& Optimization} {\bf 23} (1994), 153-167.}

\bibitem{Gunzel}\ref
{H.~G\"unzel:}
{The universal partition theorem for oriented matroids,}
{{\sl Discrete Comput.\ Geometry} {\bf 15} (1996), 121-145.}

\bibitem{Gunzel2}\ref
{H.~G\"unzel:}
{On the universal partition theorem for $4$-polytopes,}
{{\sl Discrete Comput.\ Geometry} {\bf 19} (1996), 521-552.}

\bibitem{GSL}\ref
{G.~Gonzalez-Sprinberg \& G.~Laffaille:}  
{Sur les arrangements simples de huit droites dans $\R{\rm P}^2$,}
{{\sl C.R.~Acad.~Sci.~Paris} {\bf 309} (1989), Ser.~I, 341-344.}
  
\bibitem{Gelfand}\ref 
{I.~M.~Gel'fand:}  
{General theory of hypergeometric functions,}
{{\sl Soviet Math.\ Doklady} {\bf 33} (1986), 573-577.}
 
\bibitem{GGMS}\ref 
{I. M. Gel'fand, R. M. Goresky, R. D. MacPherson \& V. Serganova: }  
{Combinatorial geometries, con\-vex polyhedra and Schubert cells,} 
{{\sl Advances in Math.} {\bf 63} (1987), 301-316.}
 
\bibitem{GKZ}\ref 
{I.~M.~Gel'fand, M.~M.~Kapranov \& A.~V.~Zelevinsky:} 
{Discriminants of polynomials in several variables and 
triangulations of Newton polyhedra,} 
{{\sl Leningrad Math.\ Journal} {\bf 2} (1991), 449-505.}

\bibitem{GMP1}\ref
{I.~M.~Gel'fand \& R.~D.~MacPherson:} 
{Geometry in Grassmannians and a generalization of the dilogarithm,} 
{{\sl Advances in Math.} {\bf 44} (1982), 279-312.}
 
\bibitem{GMP2}\ref
{I.~M.~Gel'fand \& R.~D.~MacPherson:} 
{A combinatorial formula for the Pontrjagin classes,} 
{{\sl Bull.~Amer.\ Math.\ Soc.} {\bf 26} (1992), 304-309.}

\bibitem{GelfandRybnikov}\ref
{I.~M.~Gel'fand \& G.~L.~Rybnikov:} 
{Algebraic and topological invariants of oriented matroids,}
{{\sl Soviet Math.\ Doklady} {\bf 40} (1990), 148-152.}
  
\bibitem{GelfandRybnikovStone}\ref
{I.~M.~Gel'fand, G.~L.~Rybnikov \& D.~A.~Stone:} 
{Projective orientations of matroids,}
{{\sl Advances in Math.} {\bf 113} (1995), 118-150.}

\bibitem{GelfandSerganova}\ref
{I.~M.~Gel'fand \& V.~V.~Serganova:} 
{On the general definition of a matroid and a greedoid,}
{{\sl Soviet Math.~Doklady} {\bf 33} (1987), 6-10.}

\bibitem{GerardsHochst}\ref
{B.~Gerards \& W.~Hochst\"attler:}
{Onion skins in oriented matroids,}
{RUTCOR Research Report 14-93, Rutgers University 1993;
Preprint 93.138, Mathematisches Institut, Universit\"at zu K\"oln 1993, 
3~pages.}

\bibitem{Goodman}\ref
{J.~E.~Goodman:}   
{Proof of a conjecture of Burr, Gr\"unbaum and Sloane,} 
{{\sl Discrete Math.} {\bf 32} (1980), 27-35.}
 
\bibitem{GP1}\ref
{J.~E.~Goodman \& R.~Pollack:}  
{On the combinatorial classification of non-degenerate 
configurations in the plane,} 
{{\sl J.~Combinatorial Theory} Ser.~A {\bf 29} (1980), 220-235.}
 
\bibitem{GP2}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Proof of Gr\"unbaum's conjecture on the
stretchability of certain arrangements of pseudolines,} 
{{\sl J.~Combinatorial Theory} Ser.~A {\bf 29} (1980), 385-390.}
 
\bibitem{GP3}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{A combinatorial perspective on some problems in geometry,} 
{{\sl Congressus Numerantium} {\bf 32} (1981), 383-394.}
 
\bibitem{GP4}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Three points do not determine a (pseudo-) plane,} 
{{\sl J.~Combinatorial Theory} Ser.~A {\bf 31} (1981), 215-218.}
 
\bibitem{GP5}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Helly-type theorems for pseudolines arrangements in $P^2$,} 
{{\sl J.~Combinatorial Theory} Ser.~A {\bf 32} (1982), 1-19.}
 
\bibitem{GP6}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{A theorem of ordered duality,} 
{{\sl Geometriae Dedicata} {\bf 12} (1982), 63-74.}
 
\bibitem{GP7}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Convexity theorems for generalized planar configurations,} 
{in: {\sl Convexity and Related Combinatorial Geometry} 
(Proc.~2nd Univ.~Oklahoma Conf.), Marcel Dekker 1982, pp.~73-80.}
 
\bibitem{GP8}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Multidimensional sorting,} 
{{\sl SIAM J.~Computing} {\bf 12} (1983), 484-503.}
 
\bibitem{GP9}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{On the number of $k$-subsets of a set of $n$ points in the plane,} 
{{\sl J.~Combinatorial Theory} Ser.~A {\bf 36} (1984), 101-104.}
 
\bibitem{GP10}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Semispaces of configurations, cell complexes of arrangements,} 
{{\sl J.~Combinatorial Theory} Ser.~A {\bf 37} (1984), 257-293.}

\bibitem{GP11}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{A combinatorial version of the isotopy conjecture,} 
{in: Proc.\ Conf.\ ``Discrete Geometry and Convexity'', New York 1982,
(J.E.\ Goodman, E.\ Lutwak, J.\ Malkevitch, R.\ Pollack, eds.),
{\sl Annals of the New York Academy of Sciences} {\bf 440} (1985), 12-19.}
 
\bibitem{GP12}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Geometric sorting theory,} 
{in: Proc.\ Conf.\ ``Discrete Geometry and Convexity'', New York 1982,
(J.E.\ Goodman, E.\ Lutwak, J.\ Malkevitch, R.\ Pollack, eds.),
{\sl Annals of the New York Academy of Sciences} {\bf 440} (1985), 347-354.}
 
\bibitem{GP12a}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{The $\lambda$-matrix: a computer-oriented model for geometric configurations,}
{in: Proc.\ 3.\ Kolloq.\ ``Diskrete Geometrie'', Salzburg 1985, 119-128.}

\bibitem{GP13}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Polynomial realizations of pseudolines arrangements,} 
{{\sl Comm.~Pure Applied Math.} {\bf 38} (1985), 725-732.}
 
\bibitem{GP14}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Upper bounds for configurations and polytopes in $\R^d$,}
{{\sl Discrete Comput.~Geometry} {\bf 1} (1986), 219-227.}
 
\bibitem{GP15}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{There are asymptotically far fewer polytopes than we thought,}
{{\sl Bulletin Amer.~Math.~Soc.} {\bf 14} (1986), 127-129.}

\bibitem{GP16}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{Hadwiger's transversal theorem in higher dimensions,}
{{\sl Journal Amer.\ Math.\ Soc.} {\bf 1} (1988), 301-309.}
 
\bibitem{GP17}\ref
{J.~E.~Goodman \& R.~Pollack:}   
{New bounds on higher dimensional configurations and polytopes,} 
{in: Proc.\ Third Int.\ Conf.\ {\sl Combinatorial Mathematics,}
(G.S.\ Bloom, R.L.\ Graham, J.\ Malkevitch, eds.),
{\sl Annals of the New York Academy of Sciences} {\bf 555} (1989), 205-212.}

\bibitem{GP18}\ref
{J.~E.~Goodman \& R.~Pollack:} 
{The complexity of point configurations,}
{{\sl Discrete Appl.\ Math.} {\bf 31} (1991), 167-180.}

\bibitem{GP19}\ref
{J.~E.~Goodman \& R.~Pollack:} 
{Allowable sequences and order types in discrete and computational geometry,} 
{in: ``New Trends in Discrete and Computational Geometry'' (J.~Pach, ed.),
{\sl Algorithms and Combinatorics} {\bf 10}, 
Springer-Verlag, Berlin Heidelberg 1993, 103-134}

\bibitem{GPSturmfels1}\ref
{J.~E.~Goodman, R.~Pollack \& B.~Sturmfels:}   
{Coordinate representation of order types requires exponential storage,}
{{\sl Proceedings of the 21st Annual
ACM Symposium on Theory of Computing}, Seattle 1989, 405-410.}
  
\bibitem{GPSturmfels2}\ref
{J.~E.~Goodman, R.~Pollack \& B.~Sturmfels:}  
{The intrinsic spread of a configuration in $\R^d$,}
{{\sl Journal Amer.\ Math.\ Soc.} {\bf 3} (1990), 639-651.}
 
\bibitem{GPWenger}\ref
{J.~E.~Goodman, R.~Pollack \& R.~Wenger:} 
{Geometric transversal theory,}
{in: ``New Trends in Discrete and Computational Geometry'' (J.~Pach, ed.),
{\sl Algorithms and Combinatorics} {\bf 10}, 
Springer-Verlag, Berlin Heidelberg 1993, 163-198.}

\bibitem{GPWengerZ1}\ref
{J.~E.~Goodman, R.~Pollack, R.~Wenger \& T.~Zamfirescu:}
{Every arrangement extends to a spread,}
{in {\sl Proc. Third Annual Canadian Conference on Computational Geometry}, 1991, pp.~191-194.}

\bibitem{GPWengerZ2}\ref
{J.~E.~Goodman, R.~Pollack, R.~Wenger \& T.~Zamfirescu:}
{Every arrangement extends to a spread,}
{{\sl Combinatorica} {\bf 14} (1994), 301-306.}

\bibitem{GPWengerZ3}\ref
{J.~E.~Goodman, R.~Pollack, R.~Wenger \& T.~Zamfirescu:}
{There is a universal topological plane,}
{In: {\sl Proc.\ Eighth Annual ACM Symp.\ Computational Geometry}, Berlin,
June 1992, pp.~171-176.}

\bibitem{GPWengerZ4}\ref
{J.~E.~Goodman, R.~Pollack, R.~Wenger \& T.~Zamfirescu:}
{Arrangements and topological planes,}
{{\sl American Math.\ Monthly} {\bf 101} (1994), 866-878.}

\bibitem{GPWengerZ5}\ref
{J.~E.~Goodman, R.~Pollack, R.~Wenger \& T.~Zamfirescu:}
{There are uncountably many universal topological planes,}
{{\sl Geometriae Dedicata} {\bf 59} (1996), 157-162.}

\bibitem{Greene}\ref
{C.~Greene:}   
{Acyclic orientations (Notes),} 
{in: {\sl Higher Combinatorics} (M.~Aigner, ed.), 
Reidel, Dordrecht 1977, 65-68.}
 
\bibitem{GreeneZaslawsky}\ref
{C.~Greene \& T.~Zaslavsky:}   
{On the interpretation of Whitney numbers through arrangements of hyperplanes, 
zonotopes, non-Radon partitions and orientations of graphs,} 
{{\sl Transactions Amer.\ Math.\ Soc.} {\bf 280} (1983), 97-126.}

\bibitem{GrigVor}\ref 
{D.Y.~Grigor'ev \& N.~N.~Vorobjov:}  
{Solving systems of polynomial equations in subexponential time,} 
{{\sl J.\ Symbolic Computation} {\bf 5} (1988), 37-64.}
 
\bibitem{GritzmannKlee}\ref 
{P.~Gritzmann \& V.~Klee:}  
{Computational aspects of zonotopes and their polars,} 
{in preparation.}
 
\bibitem{GritzmannSturmfels}\ref 
{P.~Gritzmann \& B.~Sturmfels:} 
{Minkowski addition of polytopes: Computational complexity and applications 
to Gr\"obner bases,} 
{{\sl SIAM J.\ Discrete Math.} {\bf 6} (1993), 246-269.}
 
\bibitem{GLS}\ref 
{M.~Gr\"otschel, L.~Lov\'asz \& A.~Schrijver:} 
{Geometric Algorithms and Combinatorial Optimization,}
{{\sl Algorithms and Combinatorics} {\bf 2}, Springer 1988.}
 
\bibitem{Grunbaum1}\ref
{B.~Gr\"unbaum:}  
{Convex Polytopes,} 
{Interscience Publ., London 1967.}
 
\bibitem{Grunbaum2}\ref
{B.~Gr\"unbaum:}  
{The importance of being straight,}
{in: Proc.\ {\sl 12th Biannual Intern.\ Seminar of the 
Canadian Math.\ Congress} (Vancouver 1969), 1970, 243-254.}
 
\bibitem{Grunbaum3}\ref
{B.~Gr\"unbaum:}   
{Arrangements and Spreads,} 
{{\sl CBMS Regional Conference Series in Math.}
{\bf 10}, Amer.\ Math.\ Soc.\ 1972.}
 
\bibitem{GrunbaumSr}\ref 
{B.~Gr\"unbaum \& V.~Sreedharan:} 
{An enumeration of simplicial $4$-polytopes with $8$ vertices,}
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\bibitem{GdO1}\ref
{A.~Guedes de Oliveira:} 
{Proje\c cc\~ao paralela em matroides orientados,}
{M.Sc.~Thesis, University of Porto, Portugal 1988.}
 
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{A.~Guedes de Oliveira:} 
{Oriented matroids and projective invariant theory,}
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{A.~Guedes de Oliveira:} 
{Oriented matroids: An essentially topological algebraic model,}
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{A.~Guedes de Oliveira:} 
{On the Steinitz exchange lemma,}
{{\sl Discrete Math.} {\bf 137} (1995), 367-370.}

\bibitem{GSS}\ref
{L.~Guibas, D.~Salesin \& J.~Stolfi:} 
{Epsilon geometry: building robust algorithms from imprecise computations,} 
{{\sl Proc.\ Fifth Annual ACM Symposium on Computational Geometry} 
1989, pp.~208-217.}
 
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{On $n$-ordered sets and order completeness,} 
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{Zonotopal complexes on the $d$-cube,}
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{Y.~O.~Hamidoune \& M.~Las Vergnas:}   
{Jeux de commutation orient\'es sur les graphes et les matro\"\i des,} 
{{\sl C.R.~Acad.~Sci.~Paris}, Ser.~A {\bf 298} (1984), 497-499.}
 
\bibitem{HamLV2}\ref
{Y.~O.~Hamidoune \& M.~Las Vergnas:}   
{Directed switching games on graphs and matroids,} 
{{\sl J.~Combinatorial Theory}, Ser.~B {\bf 40} (1986), 237-269.}
 
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{in: {\sl Topology and Computer Science} (1987), pp.~535-545.}
 
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{A characterization of oriented matroids in terms of topes,} 
{{\sl European J.~Combinatorics} {\bf 11} (1990), 41-45.}

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{Topes of oriented matroids and related structures,}
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{On conditions for an acycloid to be matroidal,}
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{Braids and Coverings: Selected Topics,}
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{Zur algorithmischen Behandlung des Steinitz-Problems,}
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{Two-colorings of simple arrangements,} 
{in: Proc.~6th Hungarian 
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{Some simple arrangements of pseudolines with a maximum number of triangles,}
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{Max-balanced flows of oriented matroids,} 
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{Geometry of Coxeter Groups,} 
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{Shellability of oriented matroids,}
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{A lattice theoretic characterization of oriented matroids,}
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{Seitenfl\"achenverb\"ande orientierter Matroide,}
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{Nested cones and onion skins,}
{{\sl Applied Math.\ Letters} {\bf 6} (1993), 67-69.}

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{A note on the weak zone theorem,}
{{\sl Congressus Numerantium} {\bf 98} (1993), 95-103.}

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{A non-visiting path, nested cones and onion skins,}
{Report 92-126, Mathematisches Institut, Universit\"at zu K\"oln 1992, 8~pages.}

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{Oriented matroids from wild spheres,}
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{Adjoints and duals of matroids linearly representable over a skewfield,}
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{A pseudoconfiguration of points without adjoint,}
{{\sl J. Combinatorial Theory}, Ser.~B, {\bf 68} (1996), 277-294.}

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{Cayley embeddings, lifting subdivisions and the Bohne-Dress theorem,}
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{Introduction to Lie algebras and representation theory,} 
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{Reflection groups and Coxeter groups,}
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\bibitem{Ingleton}\ref
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{Orientierung und Ordnungsfunktionen in kombinatorischen Geometrien,}
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{Oriented matroids in terms of order functions,}
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{A simple way to tell a simple polytope from its graph,}
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\bibitem{KlafszkyTerlaky2}\ref
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{Remarks on the feasibility problem of oriented matroids,}
{Annales Universitatis Scientiarum Budapestiensis 
de Rolando E\"otv\"os nominatae, Sectio Computatorica, 
{\bf 7} (1987), 155-157.}
 
\bibitem{KlafszkyTerlaky2a}\ref
{E.~Klafszky \& T.~Terlaky:} 
{Oriented matroids, quadratic programming and the criss-cross method,}
{(in Hungarian) {\sl Alkalmazott Mat.\ Lapok} {\bf 14} (1989), 365-375.}

\bibitem{KlafszkyTerlaky3}\ref
{E.~Klafszky \& T.~Terlaky:} 
{Some generalizations of the criss-cross method for the
linear complementarity problem of oriented matroids,} 
{{\sl Combinatorica} {\bf 9} (1989), 189-198.}
  
\bibitem{KlafszkyTerlaky4}\ref
{E.~Klafszky \& T.~Terlaky:} 
{Some generalizations of the criss-cross method for 
quadratic programming,}
{{\sl Optimization} {\bf 24} (1992), 127-139.} 

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{V.~Klee \& P.~Kleinschmidt:} 
{Convex polytopes and related complexes,}
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North-Holland/Elsevier, Amsterdam 1995, pp.~875--917.}
 
\bibitem{KleimanLaksov}\ref
{S.~Kleiman \& D.~Laksov:}  
{Schubert calculus,}
{{\sl Amer.~Math.~Monthly} {\bf 79} (1972), 1061-1082.}
 
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{P.~Kleinschmidt:} 
{On facets with non-arbitrary shapes,}
{{\sl Pacific J.\ Math.} {\bf 65} (1976), 511-515.}
 
\bibitem{Kleinschmidt2}\ref 
{P.~Kleinschmidt:} 
{Sph\"aren mit wenigen Ecken,}
{{\sl Geometriae Dedicata} {\bf 5} (1976), 307-320.}

\bibitem{KleinschmidtOnn}\ref 
{P.~Kleinschmidt \& S.~Onn:} 
{Signable posets and partitionable simplicial complexes,}
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\bibitem{KlinTZ}\ref
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{$2D$-configurations and clique-cyclic orientations of the graphs $L(K_p)$,}
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{D.~E.~Knuth:} 
{Axioms and Hulls,} 
{{\sl Lecture Notes in Computer Science} {\bf 606}, Springer 1992.}
  
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{Representation of data by pseudoline arrangements,}
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{Every simplicial polytope with at most $d{+}4$ vertices 
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{Adjoints, Schiefk\"orper und algebraische Matroide,}
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{On sets projectively equivalent to the vertices of a convex polytope,} 
{{\sl Bulletin London Math.~Soc.} {\bf 4} (1972), 6-12.}

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{{\sl C.R.~Acad.~Sci.~Paris}, Ser.A {\bf 280} (1975), 61-64.}
 
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{M.~Las Vergnas:}  
{Coordinatizable strong maps of matroids,}
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{Sur les extensions principales d'un matro\"\i de,}
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\bibitem{LasVergnas5}\ref
{M.~Las Vergnas:}   
{Acyclic and totally cyclic orientations of combinatorial geometries,}
{{\sl Discrete Math.} {\bf 20} (1977), 51-61.}
 
\bibitem{LasVergnas6}\ref
{M.~Las Vergnas:}   
{Bases in oriented matroids,} 
{{\sl J.~Combinatorial Theory}, Ser.~B {\bf 25} (1978), 283-289.}
 
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{M.~Las Vergnas:}  
{Convexity in oriented matroids,} 
{{\sl J.~Combinatorial Theory}, Ser.~B {\bf 29} (1980), 231-243.}
 
\bibitem{LasVergnas10}\ref
{M.~Las Vergnas:}   
{On the Tutte polynomial of a morphism of matroids,} 
{{\sl Ann.\ Discrete Math.} {\bf 8} (1980), 7-20.}
 
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{A correspondence between spanning trees and orientations in graphs,} 
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{M.~Las Vergnas:}   
{Order properties of lines in the plane and a conjecture of G.~Ringel,} 
{{\sl J.~Combinatorial Theory} Ser.~B {\bf 41} (1986), 246-249.}
 
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{Hamilton paths in tournaments and a problem of McMullen
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{M.~Las Vergnas:}  
{Acyclic reorientations of weakly oriented matroids,} 
{{\sl J.\ Combinatorial Theory} Ser.~B {\bf 49} (1990), 195-199.}
   
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{Oriented matroids,}
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{{\sl Linear Algebra Appl.} {\bf 48} (1982), 1-12.}
 
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{{\sl Pacific J.~Math.} {\bf 104} (1983), 155-173.}
 
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{Subspaces with well-scaled frames,} 
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{The incidence structure of subspaces with well-scaled frames,}
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{Unshellable triangulations of spheres,}
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{Arrangements of oriented hyperplanes,}
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{Spacial graphs and oriented matroids: the trefoil,}
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{{\sl Proceedings Amer.\ Math.\ Soc.} {\bf 107} (1989), 555-559.}
 
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{J.~Richter-Gebert:} 
{Non-euclidean uniform oriented matroids have bi-quadratic final polynomials,}
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{J.~Richter-Gebert:}
{On the Realizability Problem for Combinatorial Geometries -- Decision Methods,}
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{J.~Richter-Gebert:}
{Construction Methods for Oriented Matroids,}
{Dissertation, KTH Stockholm 1992.}

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{Oriented matroids with few mutations,}
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{\sl Discrete Comput.\ Geometry} {\bf 10} (1993), 251-269.}

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{Combinatorial obstructions to the lifting of weaving diagrams,}
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{Mechanical theorem proving in projective geometry,}
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{Realization Spaces of Polytopes,}
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{Testing orientability for matroids is NP-complete,}
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