## Euclideaness and final polynomials in oriented matroid theory (1993)

Venue: | COMBINATORICA |

### BibTeX

@MISC{Richter-Gebert93euclideanessand,

author = {Jürgen Richter-Gebert},

title = {Euclideaness and final polynomials in oriented matroid theory},

year = {1993}

}

### OpenURL

### Abstract

This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programms in non-euclidean oriented matroids.

### Citations

136 |
Oriented Matroids
- Björner, Vergnas, et al.
- 1993
(Show Context)
Citation Context ...enerate cycling for an oriented matroid is a method to prove its non-realizabilty. In this chapter a brief outline of oriented matroid programming will be given. For details the reader is referred to =-=[1]-=-, [2], [6]. For the rest of this chapter we assume that χ is an oriented matroid of rank d on the set E := En ∪ {f, g} without loops, coloops and parallel elements. We assume that we have a linear ord... |

23 |
Stretchability of pseudolines is NP-hard. In Applied geometry and discrete mathematics, volume 4
- Shor
- 1991
(Show Context)
Citation Context ...l LP-problem lies in O(n 6d ); here d denotes the rank and n the number of elements of χ (compare [8]). On the other hand the realization problem for oriented matroids is known to be NP-hard (compare =-=[9]-=-), so it is very unlikely that the above method works in general. Nevertheless up to now, no non-realizable oriented matroid is known, that does not admit a bi-quadratic final polynomial. Another way ... |

22 | Topology of oriented matroids - Mandel - 1981 |

20 |
Oriented matroid programming
- Fukuda
- 1982
(Show Context)
Citation Context ...roids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel =-=[6]-=-, [7] concerning non-degenerate cycling of linear programms in non-euclidean oriented matroids. 1. Introduction One of the most important problems in oriented matroid theory is to find good algorithms... |

18 |
A combinatorial abstraction of linear programming
- Bland
- 1977
(Show Context)
Citation Context ...nown, that does not admit a bi-quadratic final polynomial. Another way of finding non-realizability proofs for oriented matroids comes from the oriented matroid version of linear programming (compare =-=[2]-=-). Fukuda gave an example of an oriented matroid program that admits a cycling sequence of strictly increasing pivot steps [6]. This cannot happen in the realizable case since in this case. So finding... |

15 |
Computational Synthetic Geometry
- Bokowski, Sturmfels
- 1989
(Show Context)
Citation Context ...r deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], =-=[5]-=- for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programms in non-euclidean oriented matroids. 1. I... |

9 |
On the finding of final polynomials
- Bokowski, Richter-Gebert
- 1990
(Show Context)
Citation Context ... paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial =-=[3]-=-, [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programms in non-euclidean oriented matroids.... |

7 |
On the realizability problem of combinatorial geometries decision methods
- Richter-Gebert
- 1992
(Show Context)
Citation Context ...ic step a LP-problem has to be solved. The number of arithmetic operations needed to solve this special LP-problem lies in O(n 6d ); here d denotes the rank and n the number of elements of χ (compare =-=[8]-=-). On the other hand the realization problem for oriented matroids is known to be NP-hard (compare [9]), so it is very unlikely that the above method works in general. Nevertheless up to now, no non-r... |

3 |
On the classification of non-realizable oriented matroids, part 1: Generation
- Bokowski, Richter-Gebert
- 1990
(Show Context)
Citation Context ... provide a link between two different approaches to the above problem: the geometric approach of Edmonds, Mandel and Fukuda [6], [7] and the algebraic approach of Bokowski, Richter and Sturmfels [3], =-=[4]-=-, [5]. One algorithmic way to derive a non-realizability proof for an oriented matroid χ is to decide, whether it admits a bi-quadratic final polynomial; if so, χ is not realizable [3]. In fact this w... |