Semi-inverted linear spaces and an analogue of the broken circuit complex
Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 645-661.

The image of a linear space under inversion of some coordinates is an affine variety whose structure is governed by an underlying hyperplane arrangement. In this paper, we generalize work by Proudfoot and Speyer to show that circuit polynomials form a universal Gröbner basis for the ideal of polynomials vanishing on this variety. The proof relies on degenerations to the Stanley–Reisner ideal of a simplicial complex determined by the underlying matroid, which is closely related to the external activity complex defined by Ardila and Boocher. If the linear space is real, then the semi-inverted linear space is also an example of a hyperbolic variety, meaning that all of its intersection points with a large family of linear spaces are real.

Received: 2018-04-16
Revised: 2019-02-01
Accepted: 2019-02-12
Published online: 2019-08-01
DOI: https://doi.org/10.5802/alco.65
Keywords: Matroid, hyperplane arrangement, simplicial complex, reciprocal linear space
@article{ALCO_2019__2_4_645_0,
     author = {Scholten, Georgy and Vinzant, Cynthia},
     title = {Semi-inverted linear spaces and an analogue of the broken circuit complex},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {4},
     year = {2019},
     pages = {645-661},
     doi = {10.5802/alco.65},
     mrnumber = {3997516},
     zbl = {1417.05254},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2019__2_4_645_0/}
}
Scholten, Georgy; Vinzant, Cynthia. Semi-inverted linear spaces and an analogue of the broken circuit complex. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 645-661. doi : 10.5802/alco.65. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_645_0/

[1] Ardila, Federico; Boocher, Adam The closure of a linear space in a product of lines, J. Algebraic Combin., Volume 43 (2016) no. 1, pp. 199-235 | Article | MR 3439307 | Zbl 1331.05051

[2] Ardila, Federico; Castillo, Federico; Samper, José Alejandro The topology of the external activity complex of a matroid, Electron. J. Combin., Volume 23 (2016) no. 3, 20 pages | MR 3558045 | Zbl 1344.05036

[3] Björner, Anders; Wachs, Michelle L. Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc., Volume 349 (1997) no. 10, pp. 3945-3975 | Article | MR 1401765 | Zbl 0886.05126

[4] De Loera, Jesús A.; Sturmfels, Bernd; Vinzant, Cynthia The central curve in linear programming, Found. Comput. Math., Volume 12 (2012) no. 4, pp. 509-540 | Article | MR 2946462 | Zbl 1254.90108

[5] Fink, Alex; Speyer, David E.; Woo, Alexander A Gröbner basis for the graph of the reciprocal plane (2017) (https://arxiv.org/abs/1703.05967 )

[6] Huh, June; Wang, Botong Enumeration of points, lines, planes, etc., Acta Math., Volume 218 (2017) no. 2, pp. 297-317 | Article | MR 3733101 | Zbl 1386.05021

[7] Kummer, Mario; Vinzant, Cynthia The Chow form of a reciprocal linear space (2016) (https://arxiv.org/abs/1610.04584 ) | Zbl 07155051

[8] Maclagan, Diane; Sturmfels, Bernd Introduction to tropical geometry, Graduate Studies in Mathematics, Volume 161, American Mathematical Society, Providence, RI, 2015, xii+363 pages | MR 3287221 | Zbl 1321.14048

[9] Michałek, Mateusz; Sturmfels, Bernd; Uhler, Caroline; Zwiernik, Piotr Exponential varieties, Proc. Lond. Math. Soc. (3), Volume 112 (2016) no. 1, pp. 27-56 | Article | MR 3458144 | Zbl 1345.14048

[10] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, Volume 227, Springer-Verlag, New York, 2005, xiv+417 pages | MR 2110098 | Zbl 1090.13001

[11] Oxley, James Matroid theory, Oxford Graduate Texts in Mathematics, Volume 21, Oxford University Press, Oxford, 2011, xiv+684 pages | Article | MR 2849819 | Zbl 1254.05002

[12] Proudfoot, Nicholas; Speyer, David A broken circuit ring, Beiträge Algebra Geom., Volume 47 (2006) no. 1, pp. 161-166 | MR 2246531 | Zbl 1095.13024

[13] Proudfoot, Nicholas; Xu, Yuan; Young, Ben The Z-polynomial of a matroid, Electron. J. Combin., Volume 25 (2018) no. 1, 21 pages | MR 3785005 | Zbl 1380.05022

[14] Sanyal, Raman; Sturmfels, Bernd; Vinzant, Cynthia The entropic discriminant, Adv. Math., Volume 244 (2013), pp. 678-707 | Article | MR 3077886 | Zbl 1284.15006

[15] Shamovich, E.; Vinnikov, V. Livsic-type determinantal representations and hyperbolicity, Adv. Math., Volume 329 (2018), pp. 487-522 | Article | MR 3783420 | Zbl 1391.32009

[16] Stanley, Richard P. Combinatorics and commutative algebra, Progress in Mathematics, Volume 41, Birkhäuser Boston, Inc., Boston, MA, 1996, x+164 pages | MR 1453579 | Zbl 0838.13008

[17] Sturmfels, Bernd Gröbner bases and convex polytopes, University Lecture Series, Volume 8, American Mathematical Society, Providence, RI, 1996, xii+162 pages | MR 1363949 | Zbl 0856.13020

[18] Terao, Hiroaki Algebras generated by reciprocals of linear forms, J. Algebra, Volume 250 (2002) no. 2, pp. 549-558 | Article | MR 1899865 | Zbl 1049.13011

[19] Varchenko, A. Critical points of the product of powers of linear functions and families of bases of singular vectors, Compositio Math., Volume 97 (1995) no. 3, pp. 385-401 | Numdam | MR 1353281 | Zbl 0842.17044

[20] Wagner, David G. Multivariate stable polynomials: theory and applications, Bull. Amer. Math. Soc. (N.S.), Volume 48 (2011) no. 1, pp. 53-84 | Article | MR 2738906 | Zbl 1207.32006