## Oriented Matroids (1997)

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Venue: | IN HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Richter-Gebert97orientedmatroids,

author = {Jürgen Richter-Gebert and Günter M. Ziegler},

title = {Oriented Matroids},

booktitle = {IN HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY},

year = {1997},

pages = {111--132},

publisher = {CRC Press}

}

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### Abstract

### Citations

482 |
Matroid theory
- Oxley
- 1992
(Show Context)
Citation Context ..., +1}. Chirotope of X: The map # X : E d # {-, 0, +} (# 1 , . . . , # d ) ## sign(det(x #1 , . . . , x #d )). Ordinary (unoriented) matroids, as introduced in 1935 by Whitney (see Kung [Kun86], Oxley =-=[Oxl92]-=-), can be considered as an abstraction of vector configurations in finite dimensional vector spaces over arbitrary fields. All the bases of a matroid M have the same cardinality d, which is called the... |

197 | On the combinatorial and algebraic complexity of quanti elimination
- Basu, Pollack, et al.
- 1996
(Show Context)
Citation Context ...problem is given by the following theorem. It follows 126 J. Richter-Gebert and G.M. Ziegler from general complexity bounds for algorithmic problems about semialgebraic sets by Basu, Pollack, and Roy =-=[BPR96]-=- (see also Chapter 29 of this Handbook). THEOREM 6.3.2 Complexity of the Best General Algorithm Known The realizability of a rank d oriented matroid on n points can be decided by solving a system of S... |

183 | Lectures on Polytopes. Graduate Texts in Mathematics - Ziegler - 1994 |

122 | Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics - Ziegler - 1995 |

64 |
The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in “Topology and geometry - Rohlin seminar
- Mnëv
- 1988
(Show Context)
Citation Context ...c96b]). The universality theorem is a fundamental statement with various implications for the configuration spaces of various types of combinatorial objects. THEOREM 6.3.3 Mnev's Universality Theorem =-=[Mne88]-=- For every basic primary semialgebraic set V defined over Z there is a chirotope # of rank 3 such that V and R(#) are stably equivalent. Although some of the facts in the following list were proved ea... |

61 |
Oriented matroids
- Folkman, Lawrence
(Show Context)
Citation Context ...of sign vectors associated with the cells of the arrangement. One of the most important interpretations of oriented matroids is given by the topological representation theorem of Folkman and Lawrence =-=[FL78]-=-; see also Oriented matroids 121 [BLSWZ93, Chapters 4 and 5] and [BKMS02]. It states that oriented matroids are in bijection to (combinatorial equivalence classes of) arrangements of oriented pseudosp... |

45 |
Las Vergnas, Orientability of matroids
- Bland, M
- 1978
(Show Context)
Citation Context ...he realization space of P are stably equivalent. 6.4 SOURCES AND RELATED MATERIAL FURTHER READING The basic theory of oriented matroids was introduced in two fundamental papers, Bland and Las Vergnas =-=[BL78]-=- and Folkman and Lawrence [FL78]. We refer to the monograph by Bjorner, Las Vergnas, Sturmfels, White, and Ziegler [BLSWZ93] for a broad introduction, and for an extensive development of the theory of... |

39 | A new algorithm to find a point in every cell defined by a family of polynomials. In Quantifier elimination and cylindrical algebraic decomposition
- Basu, Pollack, et al.
- 1998
(Show Context)
Citation Context ...r bound for the worst case complexity of the realizability problem is given by the following theorem. The results follows fron general complexity results on semialgebraic set by Basu, Pollack and Roy =-=[2]-=-. THEOREM (Complexity of the best general algorithm known): The number of bit operations needed to decide the realizability of a rank d oriented matroid on n points in the Turing machine model of comp... |

36 | On the coordinatization of oriented matroids
- Bokowski, Sturmfels
- 1986
(Show Context)
Citation Context ... [BLSWZ93] for a broad introduction, and for an extensive development of the theory of oriented matroids. Other introductions and basic sources of information include Bachem and Kern [BK92], Bokowski =-=[Bok93]-=-, Bokowski and Sturmfels [BSt89a], and Ziegler [Zie95, Lect. 6 and 7]. RELATED CHAPTERS Chapter 5: Pseudoline arrangements Chapter 13: Basic properties of convex polytopes Chapter 21: Arrangements Cha... |

22 |
and W.Kern: Linear Programming Duality. An Introduction to Oriented Matroids, Universitext
- Bachem
- 1992
(Show Context)
Citation Context ...sum M(E) = M(E 1 ) # M(E 2 ) if and only if M # is a direct sum M # (E) = M # (E 1 ) #M # (E 2 ). Duality of oriented matroids captures, among other things, the concepts of linear programming duality =-=[BK92]-=- [BLSWZ93, Chap. 10] and the concept of Gale diagrams for polytopes [Gru67, Sect. 5.4] [Zie95, Lect. 6]. For the latter, we note here that the vertex set of a d-dimensional convex polytope P with d + ... |

22 | Topology of oriented matroids - Mandel - 1981 |

16 |
Axioms and Hulls, volume 606
- Knuth
- 1992
(Show Context)
Citation Context ...ment of pseudolines. (It is left as a challenging exercise to the reader to prove the nonstretchability.) Up to isomorphism this is the only simple nonstretchable arrangement of 9 pseudolines [Ric89] =-=[Knu92]-=-; every arrangement of 8 (or fewer) pseudolines is stretchable [GP80]. FIGURE 6.1.2 A nonstretchable arrangement of nine pseudolines. It was obtained by Ringel [Rin56] as a perturbation of the Pappus ... |

16 |
Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade
- Levi
- 1926
(Show Context)
Citation Context ... general, even for uniform oriented matroids of rank 3 [JMSW89]. 6.3.3 TRIANGLES AND SIMPLICIAL CELLS There is a long tradition of studying triangles in arrangements of pseudolines. In his 1926 paper =-=[Lev26]-=-, Levi already considered them to be important structures. There are good reasons for this. On the one hand, they form the simplest possible cells of full dimension, and are therefore of basic interes... |

15 |
Teilungen der Ebene durch Geraden oder topologische Geraden
- Ringel
- 1955
(Show Context)
Citation Context ...ngement of 9 pseudolines [Ric89] [Knu92]; every arrangement of 8 (or fewer) pseudolines is stretchable [GP80]. FIGURE 6.1.2 A nonstretchable arrangement of nine pseudolines. It was obtained by Ringel =-=[Rin56]-=- as a perturbation of the Pappus configuration. To associate with a projective arrangement P an oriented matroid we represent the projective plane (as customary) by the 2-sphere on which antipodal poi... |

14 |
R.: Proof of Grünbaum’s conjecture on the stretchability of certain arrangements of pseudolines
- Goodman, Pollack
- 1980
(Show Context)
Citation Context ...er to prove the nonstretchability.) Up to isomorphism this is the only simple nonstretchable arrangement of 9 pseudolines [Ric89] [Knu92]; every arrangement of 8 (or fewer) pseudolines is stretchable =-=[GP80]-=-. FIGURE 6.1.2 A nonstretchable arrangement of nine pseudolines. It was obtained by Ringel [Rin56] as a perturbation of the Pappus configuration. To associate with a projective arrangement P an orient... |

13 |
Vergnas, Convexity in oriented matroids
- Las
- 1980
(Show Context)
Citation Context ...n rank d, i.e., the regions bounded by the minimal number of d pseudohyperplanes. We call an arrangement simple if no more than d - 1 planes meet in a point. It was conjectured by Las Vergnas in 1980 =-=[L80]-=- that (as in the rank 3 case) any two simple arrangements can be transformed into each other by a sequence of flips of simplicial regions. In particular this requires that every simple arrangement con... |

13 |
Realization Spaces of Polytopes, volume 1643
- Richter-Gebert
- 1996
(Show Context)
Citation Context ...ence. Two semialgebraic sets are stably equivalent if they can be connected by a sequence of rational coordinate changes, together with certain projections with contractible fibers. (See [RGZ95], and =-=[Ric96a]-=- for details.) In particular, two stably equivalent semialgebraic sets have the same number of components, they are homotopy-equivalent, and either both or neither of them have rational points. One of... |

13 | Realization spaces of 4-polytopes are universal
- Richter-Gebert
- 1995
(Show Context)
Citation Context ...otopy equivalence. Two semialgebraic sets are stably equivalent if they can be connected by a sequence of rational coordinate changes, together with certain projections with contractible fibers. (See =-=[RGZ95]-=-, and [Ric96a] for details.) In particular, two stably equivalent semialgebraic sets have the same number of components, they are homotopy-equivalent, and either both or neither of them have rational ... |

12 |
The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes
- Altshuler, Bokowski, et al.
- 1980
(Show Context)
Citation Context ...ts known for the nonrealizable case is dramatic: SIMPLICIAL CELLS IN PSEUDOARRANGEMENTS 1. There is an arrangement of 8 pseudoplanes in rank 4 having only 7 simplicial regions (Altshuler and Bokowski =-=[ABS80]-=-, Roudne# and Sturmfels [RSt88]). 2. Every rank 4 arrangement with ns13 pseudoplanes has at least one simplicial region (Bokowski and Rohlfs [BR01]). 3. For every k > 2 there is a rank 4 arrangement o... |

12 |
Arrangements of hyperplanes
- Grünbaum
- 1971
(Show Context)
Citation Context ...he near-pencils, there are two infinite classes of simplicial line arrangements and 91 additional "sporadic" simplicial line arrangements (and many more simplicial pseudoarrangements) known =-=(Grunbaum [Gru71]-=-). 3. If P is simple, then it contains at most n(n-1) 3 triangles. For infinitely many values of n, there exists a simple arrangement with n(n-1) 3 triangles (Roudne#, Harborth). 4. Any two simple arr... |

10 |
Polarity and inner products in oriented matroids
- Billera, Munson
- 1984
(Show Context)
Citation Context ...epresent matroid polytopes have dual cell decompositions (because they are piecewise linear), but this dual cell decomposition is not in general a matroid polytope, even in rank 4 (Billera and Munson =-=[BM84]-=-; Bokowski and Schuchert [BSc95]). In other words, the order dual of the face lattice of a matroid polytope (as an abstract lattice) is not in general the face lattice of a matroid polytope. (Matroid ... |

9 | Topology of oriented matroids - Edmonds, Mandel - 1982 |

8 |
The intrinsic spread of a configuration in R d
- Goodman, Pollack, et al.
- 1990
(Show Context)
Citation Context ...nted matroids on n elements, even uniform ones, in the integer grid {1, 2, . . . , f(n)} 3 , the "coordinate size" function f(n) has to grow doubly exponentially in n (Goodman, Pollack, and =-=Sturmfels [GPS90]-=-). 7. The isotopy problem for oriented matroids (Can one given realization of M be continuously deformed, through realizations, to another given one?) has a negative solution in general, even for unif... |

7 | On the Folkman-Lawrence topological representation theorem for oriented matroids of rank 3
- Bokowski, Mock, et al.
- 1999
(Show Context)
Citation Context ...icitly while drawing Figure 6.1.2.) For a recent and reasonably elementary proof of the fact that rank 3 oriented matroids are equivalent to arrangements of pseudolines see Bokowski, Mock and Streinu =-=[BMS01]-=-. By means of this equivalence, all problems concerning pseudoline arrangements can be translated to the language of oriented matroids. For instance, the problem of stretchability is equivalent to the... |

6 |
A source book in matroid theory
- Kung
- 1986
(Show Context)
Citation Context ...ments of {-1, 0, +1}. Chirotope of X: The map # X : E d # {-, 0, +} (# 1 , . . . , # d ) ## sign(det(x #1 , . . . , x #d )). Ordinary (unoriented) matroids, as introduced in 1935 by Whitney (see Kung =-=[Kun86]-=-, Oxley [Oxl92]), can be considered as an abstraction of vector configurations in finite dimensional vector spaces over arbitrary fields. All the bases of a matroid M have the same cardinality d, whic... |

5 |
On a mutation problem of oriented matroids
- Bokowski, Rohlfs
(Show Context)
Citation Context ...g only 7 simplicial regions (Altshuler and Bokowski [ABS80], Roudne# and Sturmfels [RSt88]). 2. Every rank 4 arrangement with ns13 pseudoplanes has at least one simplicial region (Bokowski and Rohlfs =-=[BR01]-=-). 3. For every k > 2 there is a rank 4 arrangement of 4k pseudoplanes having only 3k +1 simplicial regions. (This result of Richter-Gebert [Ric93] was improved Oriented matroids 129 by Bokowski and R... |

5 |
Kombinatorische Realisierbarkeitskriterien für orientierte Matroide. Master’s thesis, Technische Hochschule
- Richter
- 1988
(Show Context)
Citation Context ...rojective plane is a collection of pseudolines such that any two pseudolines intersect in exactly one point, where they 116 J. Richter-Gebert and G.M. Ziegler cross. (See Grunbaum [Gru72] and Richter =-=[Ric89]-=-.) We will always assume that P is essential, i.e., that the intersection of all the pseudolines p i is empty. An arrangement of pseudolines behaves in many respects just like an arrangement of n line... |

5 |
Oriented matroids with few mutations
- Richter-Gebert
- 1993
(Show Context)
Citation Context ...s at least one simplicial region (Bokowski and Rohlfs [BR01]). 3. For every k > 2 there is a rank 4 arrangement of 4k pseudoplanes having only 3k +1 simplicial regions. (This result of Richter-Gebert =-=[Ric93]-=- was improved Oriented matroids 129 by Bokowski and Rohlfs [BR01] to arrangements of 5k pseudoplanes with 7k - c simplicial regions.) 4. There is a rank 4 arrangement consisting of 20 pseudoplanes for... |

5 |
Isotopic but not rigidly isotopic plane systems of straight lines
- Suvorov
- 1988
(Show Context)
Citation Context ...id on 9 elements that has no realization with rational coordinates (Perles [Gru67, p. 93]). . There is a realizable rank 3 oriented matroid on 14 elements with disconnected realization space (Suvorov =-=[Suv88]-=-; see also Richter-Gebert [Ric96b]). The universality theorem is a fundamental statement with various implications for the configuration spaces of various types of combinatorial objects. THEOREM 6.3.3... |

4 |
Uniform oriented matroids without the isotopy property, Discrete Comput
- Jaggi, Mani-Levitska, et al.
- 1989
(Show Context)
Citation Context ...iented matroids (Can one given realization of M be continuously deformed, through realizations, to another given one?) has a negative solution in general, even for uniform oriented matroids of rank 3 =-=[JMSW89]-=-. 6.3.3 TRIANGLES AND SIMPLICIAL CELLS There is a long tradition of studying triangles in arrangements of pseudolines. In his 1926 paper [Lev26], Levi already considered them to be important structure... |

4 | Two interesting oriented matroids
- Richter-Gebert
(Show Context)
Citation Context ...ization with rational coordinates (Perles [Gru67, p. 93]). . There is a realizable rank 3 oriented matroid on 14 elements with disconnected realization space (Suvorov [Suv88]; see also Richter-Gebert =-=[Ric96b]-=-). The universality theorem is a fundamental statement with various implications for the configuration spaces of various types of combinatorial objects. THEOREM 6.3.3 Mnev's Universality Theorem [Mne8... |

4 |
Simplicial cells in arrangements and mutations of oriented matroids, Geom. Dedicata 27
- Roundneff, Sturmfels
- 1988
(Show Context)
Citation Context ...case is dramatic: SIMPLICIAL CELLS IN PSEUDOARRANGEMENTS 1. There is an arrangement of 8 pseudoplanes in rank 4 having only 7 simplicial regions (Altshuler and Bokowski [ABS80], Roudne# and Sturmfels =-=[RSt88]-=-). 2. Every rank 4 arrangement with ns13 pseudoplanes has at least one simplicial region (Bokowski and Rohlfs [BR01]). 3. For every k > 2 there is a rank 4 arrangement of 4k pseudoplanes having only 3... |

4 |
Mnëv: The universality theorems on the classification problem of configuration varieties and convex polytope varieties
- E
(Show Context)
Citation Context ...e (Suvorov). The Universality Theorem is a fundamental statement with various implications for the configuration spaces of various types of combinatorial objects. THEOREM (Mnëv’s Universality Theor=-=em [14]):-=- For every primary semialgebraic set V defined over Z there is a chirotope χ of rank 3 such that V and R(χ) are stably equivalent. Although some of the facts in the following list were proved earlie... |

3 |
Arrangements and Spreads, volume 10
- Grünbaum
- 1972
(Show Context)
Citation Context ...pseudolines in the projective plane is a collection of pseudolines such that any two pseudolines intersect in exactly one point, where they 116 J. Richter-Gebert and G.M. Ziegler cross. (See Grunbaum =-=[Gru72]-=- and Richter [Ric89].) We will always assume that P is essential, i.e., that the intersection of all the pseudolines p i is empty. An arrangement of pseudolines behaves in many respects just like an a... |

3 |
Knuth: Axioms and Hulls
- E
- 1992
(Show Context)
Citation Context ...allenging exercise to the reader to prove the non-stretchability of this particular pseudoline arrangement. Up to isomorphism this is the only simple non-stretchable arrangement of 9 pseudolines [16] =-=[11]-=-; every arrangement of 8 (or fewer) pseudolines is stretchable. Figure 7.2 A non-stretchable arrangement of nine pseudolines. To associate with a projective arrangement P an oriented matroid we repres... |

2 |
Neighborly 6-polytopes with 10 vertices
- Bokowski, Shemer
- 1987
(Show Context)
Citation Context ...ssfully completed for the classification of all simplicial 3-spheres with 9 vertices (Altshuler, Bokowski, and Steinberg [ABS80]) and of all neighborly 5-spheres with 10 vertices (Bokowski and Shemer =-=[BSh87]-=-) into polytopes and nonpolytopes. At the core of the matroidal approach lies the following hierarchy: # simplicial spheres # -# # uniform matroid polytopes # -# # convex polytopes # . The plan of att... |

2 |
Computational Synthetic Geometry. Volume 1355
- Bokowski, Sturmfels
- 1989
(Show Context)
Citation Context ...of basis orientations. Here an algebraic connection to determinant identities becomes obvious. Chirotopes are the main tool for translating problems in oriented matroid theory to an algebraic setting =-=[BSt89a]-=-. They also form a description of oriented matroids that is very practical for many algorithmic purposes (for instance in computational geometry; see Knuth [Knu92]). Definition: Let E := {1, . . . , n... |

2 |
An infinite family of minor-minimal nonrealizable 3-chirotopes, Mathematische Zeitschrift 200
- Bokowski, Sturmfels
- 1989
(Show Context)
Citation Context ...troid whose realization space is homotopy-equivalent to #. 5. Realizability of rank 3 oriented matroids cannot be characterized by excluding a finite set of "forbidden minors" (Bokowski and =-=Sturmfels [BSc89b]). 6. In o-=-rder to realize all combinatorial types of integral rank 3 oriented matroids on n elements, even uniform ones, in the integer grid {1, 2, . . . , f(n)} 3 , the "coordinate size" function f(n... |

2 | Convex Polytopes, Interscience, London 1967 Second edition prepared by - GRÜNBAUM - 2003 |

2 |
Simplicial cells in arrangements of hyperplanes
- Shannon
- 1979
(Show Context)
Citation Context ... prove that any two members in this subclass can be connected by a sequence of flips of simplicial regions and that each realizable arrangement contains at least one simplicial cell. In fact, Shannon =-=[Sha79]-=- proved that every arrangement (even the nonsimple) of n projective hyperplanes in rank d contains at least n simplicial regions. More precisely, for every hyperplane h there are at least d simplices ... |

2 | Oriented matroids, Chapter 2.5 in Handbook of Convex Geometry - Bokowski - 1993 |

2 | Note on Shortest and Nearest Lattice Vectors 429/1995 Jorg Rambau - Massey, Simion, et al. - 1995 |

1 |
Altshuler's sphere M 963 revisited
- Bokowski, Schuchert
- 1995
(Show Context)
Citation Context ... dual cell decompositions (because they are piecewise linear), but this dual cell decomposition is not in general a matroid polytope, even in rank 4 (Billera and Munson [BM84]; Bokowski and Schuchert =-=[BSc95]-=-). In other words, the order dual of the face lattice of a matroid polytope (as an abstract lattice) is not in general the face lattice of a matroid polytope. (Matroid polytopes form an important tool... |

1 |
Stretchability of pseudolines is NP
- Shor
- 1991
(Show Context)
Citation Context ...THE UNIVERSALITY THEOREM 1. The full field of algebraic numbers is needed to realize all oriented matroids of rank 3. 2. The realizability problem for oriented matroids is NP-hard (Mnev [Mne88], Shor =-=[Sho91]). 3. The -=-realizability problem for oriented matroids is (polynomial time) equivalent to the "Existential Theory of the Reals" (Mnev [Mne88]). Oriented matroids 127 4. For every finite simplicial comp... |

1 |
Jürgen Richter-Gebert and Günter M
- Björner, Vergnas, et al.
- 1993
(Show Context)
Citation Context ...theory of oriented matroids was introduced in two fundamental papers, Bland & Las Vergnas [4] and Folkman & Lawrence [9]. We refer to the monograph by Björner, Las Vergnas, Sturmfels, White & Ziegler=-= [3]-=- for a broad introduction, and for an extensive development of the theory of oriented matroids with a reasonably complete bibliography. Other introductions and basic sources of information include Bac... |

1 |
Schuchert: Altshuler's sphere M
- Bokowski
- 1993
(Show Context)
Citation Context ...topes have dual cell decompositions (because they are piecewise linear), but this dual cell decomposition is not in general a matroid polytope, even in rank 4. (Billera & Munson; Bokowski & Schuchert =-=[6]-=-). In other words, the order dual of the face lattice of a matroid polytope (as an abstract lattice) is not in general the face lattice of a matroid polytope. (Matroid polytopes form an important tool... |

1 | Bokowski: Oriented matroids, Chapter 2.5 in: Handbook of Convex Geometry - unknown authors - 1993 |

1 | Oriented Matroids 23 [10] B. Gr unbaum: Arrangements and - Soc - 1972 |

1 | Las Vergnas: Convexity in oriented matroids - unknown authors - 1980 |

1 | Richter-Gebert: Realization spaces of 4-polytopes are universal, Habilitationsschrift /preprint, TU Berlin 1994/1995, 111 pages; available per WWW from http://www.math.tu-berlin.de/richter [18] G. Ringel: Teilungen der Ebene durch Geraden oder topologisch - unknown authors - 1956 |