The universality theorem on the oriented
                    matroid stratification of the space of real
                    matrices

Mnëv, Nikolai

doi:10.1090/dimacs/006/16

Cited by 9 publications

(9 citation statements)

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“…This result----even a stronger one--has been stated earlier by Mn~v [5]. At the end of this section we compare the earlier statement and the present one.…”

Section: Introductionsupporting

confidence: 87%

“…This yields a weaker theorem, replacing "diffeomorphic" by "birationally isomorphic." We emphasize that the original statements of the Universal Partition Theorem and the Universality Theorem [4], [5] go beyond the scope of the results proven here in the following main points; however, recall that at present no (complete) proof of these results is available:…”

Section: Introductionmentioning

confidence: 84%

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The universal partition theorem for oriented matroids

Günzel

1996

Discrete Comput Geom

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We study the online submodular maximization problem with free disposal under a ma-troid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k-uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches 1 α∞ ≈ 0.3178 as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015) and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k-uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1 α∞ ≈ 0.3178. Finally, our techniques can be extended to a more general problem that generalizes both the online submodular maximization problem and the online bipartite matching problem with free disposal. Using the techniques developed in this paper, we give constant-competitive algorithms for the submodular online bipartite matching problem.

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“…This result----even a stronger one--has been stated earlier by Mn~v [5]. At the end of this section we compare the earlier statement and the present one.…”

Section: Introductionsupporting

confidence: 87%

Section: Introductionmentioning

confidence: 84%

The universal partition theorem for oriented matroids

Günzel

1996

Discrete Comput Geom

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“…While line arrangements are simple objects to define, these moduli spaces can be very complicated. If one generalizes from line arrangements to hyperplane arrangements, such moduli spaces can realize any algebraic variety by Mnev's universality theorem [21]. Real and complex line arrangements have been studied for over a century, though many questions remain unanswered.…”

Section: Introductionmentioning

confidence: 99%

Topological Realizations of Line Arrangements

Ruberman

Starkston

2017

International Mathematics Research Notices

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A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines.The classic version of this would ask for lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to RP 1 ) in the real projective plane. In this paper we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, 2-spheres.

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“…As a consequence of Mnëv's Universality Theorem for oriented matroids realization spaces of polytopes can be as "complex" as arbitrary basic semialgebraic sets (see [7], [2], [8], [11], and [4]). This stands in contrast to the well-known Steinitz Theorem (see [12] and [14]), which implies that realization spaces of 3-polytopes are always topologically trivial.…”

Section: Introductionmentioning

confidence: 99%

“…Here, we follow the concept of stable equivalence introduced by Mnëv [7], [8], we especially refer to the refined concept introduced by Richter-Gebert [9], which is slightly strengthened here again. Compared with [9] our setting immediately implies that a stable projection mapping is a trivial fibration.…”

Section: Introductionmentioning

confidence: 99%

On the Universal Partition Theorem for 4-Polytopes

Günzel¹

1998

Discrete Comput Geom

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By means of sign-patterns any finite family of polynomials induces a decomposition of R n into basic semialgebraic sets. In case of integer coefficients the latter decomposition roughly appears to be a partition into realization spaces of 4-polytopes. The latter is stated by the Universal Partition Theorem for 4-polytopes by Richter-Gebert. The present paper presents a different proof. As its main tool, the von Staudt polytope is introduced. The von Staudt polytope constitutes the polytopal equivalent of the well-known von Staudt constructions for point configurations. With the aid of the von Staudt polytope the original ideas of universality theory can be directly applied to the polytopal case. Moreover, a new method for representing real values (on a computation line) by polytopal means is presented. This method implies a bundling strategy in order to duplicate the encoded information. Based on this approach, the following complexity result is obtained. The incidence code of a polytope, exhibiting a realization space equivalent to a given semialgebraic set, can be computed in the same time that it requires to generate the defining polynomial system.

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The universality theorem on the oriented matroid stratification of the space of real matrices

Cited by 9 publications

References 0 publications

The universal partition theorem for oriented matroids

The universal partition theorem for oriented matroids

Topological Realizations of Line Arrangements

On the Universal Partition Theorem for 4-Polytopes

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