The tropical critical points of an affine matroid. (English) Zbl 1542.05027
Summary: We prove that the number of tropical critical points of an affine matroid \((M,e)\) is equal to the beta invariant of \(M\). Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of \((M,e)\) and the inverted Bergman fan of \(N=(M/e)^\perp\), where \(e\) is an element of \(M\) that is neither a loop nor a coloop. Equivalently, for a generic weight vector \(w\) on \(E-e\), this is the number of ways to find weights \((0,x)\) on \(M\) and \(y\) on \(N\) with \(x+y=w\) such that, on each circuit of \(M\) (resp., \(N\)), the minimum \(x\)-weight (resp., \(y\)-weight) occurs at least twice. This answers a question of Sturmfels.
MSC:
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
| 14C15 | (Equivariant) Chow groups and rings; motives |
References:
| [1] | Agostini, D., Brysiewicz, T., Fevola, C., Kühne, L., Sturmfels, B., Telen, S., and Lam, T., Likelihood degenerations, Adv. Math., 414 (2023), 108863. · Zbl 1535.14106 |
| [2] | Ardila, F., Denham, G., and Huh, J., Lagrangian geometry of matroids, J. Amer. Math. Soc., 36 (2023), pp. 727-794. · Zbl 1512.05068 |
| [3] | Ardila, F. and Escobar, L., The harmonic polytope, Selecta Math., 27 (2021), 91. · Zbl 1476.52009 |
| [4] | Ardila, F., Fink, A., and Rincón, F., Valuations for matroid polytope subdivisions, Canad. J. Math., 62 (2010), pp. 1228-1245. · Zbl 1231.05053 |
| [5] | Adiprasito, K., Huh, J., and Katz, E., Hodge theory for combinatorial geometries, Ann. of Math. (2), 188 (2018), pp. 381-452. · Zbl 1442.14194 |
| [6] | Ardila, F. and Klivans, C. J., The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B, 96 (2006), pp. 38-49. · Zbl 1082.05021 |
| [7] | Ardila-Mantilla, F., Intersection Theory of Matroids: Variations on a Theme, preprint, arXiv:2401.07916, 2024. |
| [8] | Berget, A., Eur, C., Spink, H., and Tseng, D., Tautological classes of matroids, Invent. Math., 233 (2023), pp. 951-1039. · Zbl 07704061 |
| [9] | Björner, A., The homology and shellability of matroids and geometric lattices, Matroid Appl., 40 (1992), pp. 226-283. · Zbl 0772.05027 |
| [10] | Catanese, F., Hoşten, S., Khetan, A., and Sturmfels, B., The maximum likelihood degree, Amer. J. Math., 128 (2006), pp. 671-697. · Zbl 1123.13019 |
| [11] | Crapo, H. H., A higher invariant for matroids, J. Combin. Theory, 2 (1967), pp. 406-417. · Zbl 0168.26203 |
| [12] | Derksen, H. and Fink, A., Valuative invariants for polymatroids, Adv. Math., 225 (2010), pp. 1840-1892. · Zbl 1221.05031 |
| [13] | Eur, C., Fife, T., Samper, J. A., and Seynnaeve, T., Reciprocal maximum likelihood degrees of diagonal linear concentration models, Matematiche (Catania), 76 (2021), pp. 447-459. · Zbl 1524.13080 |
| [14] | Fulton, W. and Sturmfels, B., Intersection theory on toric varieties, Topology, 36 (1997), pp. 335-353. · Zbl 0885.14025 |
| [15] | Huh, J. and Sturmfels, B., Likelihood geometry, in Combinatorial Algebraic Geometry, , Springer, Cham, 2014, pp. 63-117. · Zbl 1328.14004 |
| [16] | Huh, J., Tropical geometry of matroids, Current Dev. Math., 2016 (2018), pp. 1-46. · Zbl 1423.14004 |
| [17] | Mikhalkin, G. and Rau, J., Tropical Geometry, Lecture Notes, 2010, https://www.math.uni-tuebingen.de/user/jora/downloads/main.pdf (Accessed 13 June 2024). |
| [18] | Maclagan, D. and Sturmfels, B., Introduction to Tropical Geometry, , American Mathematical Society, Providence, RI, 2015. · Zbl 1321.14048 |
| [19] | Oxley, J. G., Matroid Theory, , Oxford University Press, Oxford, UK, 2006. · Zbl 1100.18001 |
| [20] | Sturmfels, B., Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics 97, , American Mathematical Society, Providence, RI, 2002. · Zbl 1101.13040 |
| [21] | Sturmfels, B., personal communication, Workshop on Nonlinear Algebra and Combinatorics from Physics, Center for the Mathematical Sciences and Applications at Harvard University, 2022. |
| [22] | Sturmfels, B. and Uhler, C., Multivariate Gaussian, semidefinite matrix completion, and convex algebraic geometry, Ann. Inst. Statist. Math., 62 (2010), pp. 603-638. · Zbl 1440.62255 |
| [23] | Varchenko, A., Critical points of the product of powers of linear functions and families of bases of singular vectors, Compos. Math., 97 (1995), pp. 385-401. · Zbl 0842.17044 |
| [24] | Welsh, D. J. A., Matroid Theory, Academic Press, London, 1976. · Zbl 0343.05002 |
| [25] | Zaslavsky, T., Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes, , American Mathematical Society, Providence, RI, 1975. · Zbl 0296.50010 |
| [26] | Ziegler, G. M., Matroid shellability, \( \beta \)-systems, and affine hyperplane arrangements, J. Algebraic Combin., 1 (1992), pp. 283-300. · Zbl 0782.05022 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
