## Combining realization space models of polytopes.(English)Zbl 1509.52013

Summary: In this paper we examine four different models for the realization space of a polytope: the classical model, the Grassmannian model, the Gale transform model, and the slack variety. Respectively, they identify realizations of the polytopes with the matrix whose columns are the coordinates of their vertices, the column space of said matrix, their Gale transforms, and their slack matrices. Each model has been used to study realizations of polytopes. In this paper we establish very explicitly the maps that allow us to move between models, study their precise relationships, and combine the strengths of different viewpoints. As an illustration, we combine the compact nature of the Grassmannian model with the slack variety to obtain a reduced slack model that allows us to perform slack ideal calculations that were previously out of computational reach. These calculations allow us to answer the question of F. Criado and F. Santos [Exp. Math. 31, No. 2, 461–473 (2022; Zbl 1494.52007)], about the realizability of a family of prismatoids, in general in the negative by proving the non-realizability of one of them.

### MSC:

 52B99 Polytopes and polyhedra 52B11 $$n$$-dimensional polytopes 15B48 Positive matrices and their generalizations; cones of matrices 05E45 Combinatorial aspects of simplicial complexes 14P10 Semialgebraic sets and related spaces

Zbl 1494.52007

SlackIdeals
Full Text:

### References:

 [1] Adiprasito, KA; Ziegler, GM, Many projectively unique polytopes, Invent. Math., 199, 3, 581-652 (2015) · Zbl 1339.52011 [2] Altshuler, A.; Steinberg, L., Enumeration of the quasisimplicial $$3$$-spheres and $$4$$-polytopes with eight vertices, Pacific J. Math., 113, 2, 269-288 (1984) · Zbl 0512.52004 [3] Altshuler, A.; Steinberg, L., The complete enumeration of the $$4$$-polytopes and $$3$$-spheres with eight vertices, Pacific J. Math., 117, 1, 1-16 (1985) · Zbl 0512.52003 [4] Ardila, F.; Rincón, F.; Williams, L., Positively oriented matroids are realizable, J. Eur. Math. Soc. (JEMS), 19, 3, 815-833 (2017) · Zbl 1358.05045 [5] Bokowski, J.; Sturmfels, B., Polytopal and nonpolytopal spheres: an algorithmic approach, Israel J. Math., 57, 3, 257-271 (1987) · Zbl 0639.52004 [6] Brandt, M.; Wiebe, A., The slack realization space of a matroid, Algebr. Comb., 2, 4, 663-681 (2019) · Zbl 1420.52017 [7] Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, Cham (2015) · Zbl 1335.13001 [8] Criado, F.; Santos, F., Topological prismatoids and small simplicial spheres of large diameter, Exp. Math. (2019) · Zbl 1494.52007 [9] Dobbins, MG, Realizability of polytopes as a low rank matrix completion problem, Discrete Comput. Geom., 51, 4, 761-778 (2014) · Zbl 1310.52015 [10] Firsching, M., Realizability and inscribability for simplicial polytopes via nonlinear optimization, Math. Program., 166, 1-2, 273-295 (2017) · Zbl 1379.52017 [11] Firsching, M., The complete enumeration of $$4$$-polytopes and $$3$$-spheres with nine vertices, Israel J. Math., 240, 1, 417-441 (2020) · Zbl 1455.52008 [12] Fukuda, K.; Miyata, H.; Moriyama, S., Complete enumeration of small realizable oriented matroids, Discrete Comput. Geom., 49, 2, 359-381 (2013) · Zbl 1278.52014 [13] Gouveia, J.; Grappe, R.; Kaibel, V.; Pashkovich, K.; Robinson, RZ; Thomas, RR, Which nonnegative matrices are slack matrices?, Linear Algebra Appl., 439, 10, 2921-2933 (2013) · Zbl 1283.15103 [14] Gouveia, J.; Macchia, A.; Thomas, RR; Wiebe, A., The slack realization space of a polytope, SIAM J. Discrete Math., 33, 3, 1637-1653 (2019) · Zbl 1423.52032 [15] Gouveia, J., Macchia, A., Thomas, R.R., Wiebe, A.: Projectively unique polytopes and toric slack ideals. J. Pure Appl. Algebra 224(5), # 106229 (2020) [16] Gouveia, J.; Pashkovich, K.; Robinson, RZ; Thomas, RR, Four-dimensional polytopes of minimum positive semidefinite rank, J. Comb. Theory Ser. A, 145, 184-226 (2017) · Zbl 1360.52006 [17] Grünbaum, B., Convex Polytopes (2003), New York: Springer, New York · Zbl 1024.52001 [18] Legendre, A.M.: Éléments de Géométrie. Imprimerie Firmin Didot, Pére et Fils, Paris (1823). https://archive.org/details/lmentsdegomtrie10legegoog [19] Macchia, A., Wiebe, A.: Slack ideals in Macaulay2. In: Mathematical Software—ICMS 2020 (Braunschweig 2020), pp. 222-231. Springer, Cham (2020) · Zbl 1503.52006 [20] Maclagan, D.; Sturmfels, B., Introduction to Tropical Geometry (2015), Providence: American Mathematical Society, Providence · Zbl 1321.14048 [21] Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Topology and Geometry—Rohlin Seminar. Lecture Notes in Math., vol. 1346, pp. 527-543. Springer, Berlin (1988) · Zbl 0667.52006 [22] Mumford, D., The Red Book of Varieties and Schemes (1999), Berlin: Springer, Berlin · Zbl 0945.14001 [23] Postnikov, A.: Total positivity, Grassmannians, and networks (2006). arXiv:math/0609764 [24] Richter, J.; Sturmfels, B., On the topology and geometric construction of oriented matroids and convex polytopes, Trans. Am. Math. Soc., 325, 1, 389-412 (1991) · Zbl 0743.05011 [25] Richter-Gebert, J., Realization Spaces of Polytopes (1996), Berlin: Springer, Berlin · Zbl 0866.52009 [26] Richter-Gebert, J.; Ziegler, GM, Realization spaces of $$4$$-polytopes are universal, Bull. Am. Math. Soc., 32, 4, 403-412 (1995) · Zbl 0853.52012 [27] Steinitz, E.: Polyeder und Raumeinteilungen. In: Encyklopädie der Mathematischen Wissenschaften, vol. 3-1-2, # 12 (1922) [28] Steinitz, E., Rademacher, H.: Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie. Grundlehren der Mathematischen Wissenschaften, vol. 41. Springer, Berlin-New York (1976) · Zbl 0325.52001 [29] Wiebe, A.: Realization Spaces of Polytopes and Matroids. PhD thesis, University of Washington (2019) [30] Yannakakis, M., Expressing combinatorial optimization problems by linear programs, J. Comput. Syst. Sci., 43, 3, 441-466 (1991) · Zbl 0748.90074 [31] Zheng, H.: Ear decompostion and balanced $$2$$-neighborly simplicial manifolds. Electr. J. Comb. 27(1), # P1.10 (2020) [32] Ziegler, GM, Nonrational configurations, polytopes, and surfaces, Math. Intell., 30, 3, 36-42 (2008) · Zbl 1210.00047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.