zbMATH — the first resource for mathematics

Local tropical linear spaces. (English) Zbl 1281.14049
Tropical linear spaces were introduced by D. Speyer [SIAM J. Discrete Math. 22, No. 4, 1527–1558 (2008; Zbl 1191.14076)]. They are \(m\)-dimensional polyhedral complexes in \(\mathbb{R}^n\), closed under sum of multiples of the vector \((1,\ldots,1)\). In this paper the author studies tropical linear spaces locally. More precisely, to every tropical linear space \(L\) there corresponds a matroid \(M\). Then to each basis \(B\) of \(M\), there corresponds a subcomplex \(L_B\) of \(L\), which is the local tropical linear space determined by \(B\), by definition. Clearly \[ L=\bigcup_BL_B. \] If \(L\) is \(m\)-dimensional, the author shows that \(L_B\) is homeomorphic to Euclidean space \(\mathbb{R}^m\), via a piecewise linear map. He gives a projection map from \(\mathbb{R}^n\) onto any given tropical linear space \(L\). On the combinatorial side, the author proves that local tropical linear spaces \(L_B\) are dual to mixed subdivisions of Minkowski sums of simplices. This allows him to give upper bounds for the \(f\)-vector of \(L_B\), making a distinction between the number of \(i\)-dimensional faces that become bounded when modding out by the lineality space spanned by the vector \((1,\ldots,1)\) and the number of arbitrary \(i\)-dimensional faces. Tight upper bound for these numbers are \[ {{n-i-1}\choose {n-m-i,i-1,m-i}}={{n-i-1}\choose {i-1}}{{n-2i}\choose {m-i}} \] and \[ {{n-i-1}\choose {m-i}}{{n-1}\choose {i-1}}, \] respectively. These bounds are related to similar bounds for tropical linear spaces, conjectured by Speyer in 2008, and proved in some cases by the latter. This is the so called \(f\)-conjecture. The author finishes this very interesting and well-written paper by defining conical tropical linear spaces and giving a direct proof of the \(f\)-conjecture for them, based on the tight upper bounds he has found previously.

14T05 Tropical geometry (MSC2010)
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
Full Text: DOI arXiv
[1] Brualdi, R.A.: Transversal matroids, combinatorial geometries. Encyclopedia of Mathematics and Its Applications, vol. 29, pp. 72-97. Cambridge University Press, Cambridge (1987)
[2] Corel, E, Gérard-levelt membranes, J. Algebraic Com., 37, 757-776, (2013) · Zbl 1271.34089
[3] Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1-27 (2004) (electronic) · Zbl 1054.52004
[4] Dickenstein, A., Feichtner, E.M., Sturmfels, B.: Tropical discriminants. J. Am. Math. Soc. 20(4), 1111-1133 (2007) · Zbl 1166.14033
[5] Dress, AMW; Walter, W, Valuated matroids, Adv. Math., 93, 214-250, (1992) · Zbl 0754.05027
[6] Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and Bergman fans. Port. Math. (N. S.) 62(4), 437-468 (2005) · Zbl 1092.52006
[7] Francois, G; Rau, J, The diagonal of tropical matroid varieties and cycle intersections, Collect. Math., 64, 185-210, (2013) · Zbl 1312.14144
[8] Gelfand I.M., Goresky R.M., MacPherson, R.D., Serganova, V.V.: Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63(3), 301-316 (1987) · Zbl 0622.57014
[9] Herrmann, S., Jensen, A., Joswig, M., Sturmfels, B.: How to draw tropical planes. Electron. J. Comb. 16(2), Special volume in honor of Anders Bjorner, Research Paper 6, 26 (2009) · Zbl 1195.14080
[10] Herrmann, S., Joswig, M., Speyer D.:, Dressians, tropical Grassmannians, and their rays. Forum Math. (to appear). http://arxiv.org/abs/1112.1278 · Zbl 1308.14068
[11] Kapranov, M.M.: Chow Quotients of Grassmannians. Israel. M. Gelfand Seminar. Advances in Soviet Mathematics, vol. 16, pp. 29-110. American Mathematical Society, Providence, RI (1993) · Zbl 0811.14043
[12] Katz E., Payne, S.: Realization spaces for tropical fans. In: Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia, vol. 6, pp. 73-88. Springer, Berlin (2011) · Zbl 1248.14066
[13] Murota, K; Tamura, A, On circuit valuation of matroids, Adv. Appl. Math., 26, 192-225, (2001) · Zbl 0979.05028
[14] Postnikov, A, Permutohedra, associahedra, and beyond, Int. Math. Res. Notices, 2009, 1026-1106, (2009) · Zbl 1162.52007
[15] Rincón, F, Isotropical linear spaces and valuated delta-matroids, J. Comb. Theory A, 119, 14-32, (2012) · Zbl 1232.05040
[16] Rincón, F, Computing tropical linear spaces, J. Symb Comput., 51, 86-98, (2013) · Zbl 1319.14060
[17] Santos, F, The Cayley trick and triangulations of products of simplices, integer points in polyhedra: geometry, number theory, algebra, optimization, Contemp. Math., 374, 151-177, (2005)
[18] Shaw K.M.: A tropical intersection product in matroidal fans. Preprint (2010). http://arxiv.org/abs/1010.3967 · Zbl 1314.14113
[19] Speyer, D, Tropical linear spaces, SIAM J. Discret. Math., 22, 1527-1558, (2008) · Zbl 1191.14076
[20] Speyer, D, A matroid invariant via the \(K\)-theory of the Grassmannian, Adv. Math., 221, 882-913, (2009) · Zbl 1222.14131
[21] Speyer, D; Sturmfels, B, The tropical Grassmannian, Adv. Geom., 4, 389-411, (2004) · Zbl 1065.14071
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.