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The topology of the space of matrices of Barvinok rank two
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Kombinatorik)
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Kombinatorik)
2010 (English)In: Beiträge zur Algebra und Geometrie, ISSN 0138-4821, Vol. 51, no 2, 373-390 p.Article in journal (Refereed) Published
Abstract [en]

The Barvinok rank of a d x n matrix is the minimum number of  points in Rd such that the tropical convex hull of the points contains all columns of the matrix. The concept originated in work by Barvinok and others on the travelling salesman problem. Our object of study is the space of real d x n matrices of Barvinok rank two. Let Bd,n denote this space modulo rescaling and translation. We show that Bd,n is a manifold, thereby settling a  conjecture due to Develin. In fact, Bd,n is homeomorphic to the quotient of the product of spheres Sd-2 x Sn-2 under the involution which sends each point to its antipode simultaneously in both  components.  In addition, using discrete Morse theory, we compute the integral homology of Bd,n. Assuming d \ge n, for odd d the homology turns out to be   isomorphic to that of Sd-2 x RPn-2. This  is true also for even d up to degree d-3, but the two cases differ from degree d-2 and up. The homology computation straightforwardly extends to more general  complexes of the form (Sd-2 x X)//Z2, where X is a finite cell  complex of dimension at most d-2 admitting a free  Z2-action.

Place, publisher, year, edition, pages
Lemgo, Germany: Heldermann Verlag , 2010. Vol. 51, no 2, 373-390 p.
Keyword [en]
Barvinok rank, tropical convexity, antipodal action, discrete Morse theory
National Category
Geometry Discrete Mathematics
URN: urn:nbn:se:kth:diva-70298ScopusID: 2-s2.0-77957035353OAI: diva2:486232
Swedish Research Council Grant 2006-3279
QC 20120209Available from: 2012-01-30 Created: 2012-01-30 Last updated: 2012-02-09Bibliographically approved

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