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The slack realization space of a matroid. (English) Zbl 1420.52017
Summary: We introduce a new model for the realization space of a matroid, which is obtained from a variety defined by a saturated determinantal ideal, called the slack ideal, coming from the vertex-hyperplane incidence matrix of the matroid. This is inspired by a similar model for the slack realization space of a polytope. We show how to use these ideas to certify non-realizability of matroids, and describe an explicit relationship to the standard Grassmann-Plücker realization space model. We also exhibit a way of detecting projectively unique matroids via their slack ideals by introducing a toric ideal that can be associated to any matroid.

52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
Macaulay2; Matroids
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[1] Björner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; Ziegler, G., Oriented Matroids, (1993), Cambridge University Press · Zbl 0773.52001
[2] Bokowski, J.; Sturmfels, B., Computational synthetic geometry, 1355, vi+168 p. pp., (1989), Springer-Verlag, Berlin · Zbl 0683.05015
[3] Chen, J., Matroids: A Macaulay2 package, (2015)
[4] Cox, D. A.; Little, J.; O’Shea, D., Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, xvi+646 p. pp., (2015), Springer: Springer, Cham · Zbl 1335.13001
[5] Gordon, G.; Mcnulty, J., Matroids: a geometric introduction, xvi+393 p. pp., (2012), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1253.05002
[6] Gouveia, J.; Macchia, A.; Thomas, R. R.; Wiebe, A., The Slack Realization Space of a Polytope
[7] Gouveia, J.; Parrilo, P. A.; Thomas, R. R., Lifts of convex sets and cone factorizations, Math. Oper. Res., 38, 2, 248-264, (2013) · Zbl 1291.90172
[8] Gouveia, J.; Pashkovich, K.; Robinson, R. Z.; Thomas, R. R., Four-dimensional polytopes of minimum positive semidefinite rank, J. Combin. Theory Ser. A, 145, 184-226, (2017) · Zbl 1360.52006
[9] Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry
[10] Mnëv, N. E., Topology and geometry — Rohlin Seminar, 1346, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, 527-543, (1988), Springer: Springer, Berlin · Zbl 0667.52006
[11] Ohsugi, H.; Hibi, T., Toric ideals generated by quadratic binomials, J. Algebra, 218, 2, 509-527, (1999) · Zbl 0943.13014
[12] Oxley, J., Matroid theory, 21, xiv+684 p. pp., (2011), Oxford University Press: Oxford University Press, Oxford · Zbl 1254.05002
[13] Rothvoss, T., STOC’14 — Proceedings of the 2014 ACM Symposium on Theory of Computing, The matching polytope has exponential extension complexity, 263-272, (2014), ACM: ACM, New York · Zbl 1315.90038
[14] Villarreal, R. H., Rees algebras of edge ideals, Comm. Algebra, 23, 9, 3513-3524, (1995) · Zbl 0836.13014
[15] Yannakakis, M., Expressing combinatorial optimization problems by linear programs, J. Comput. System Sci., 43, 3, 441-466, (1991) · Zbl 0748.90074
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