# zbMATH — the first resource for mathematics

On the toric ideal of a matroid. (English) Zbl 1297.14055
Let $$M$$ be a matroid on a ground set $$E$$ with bases $$\mathcal B$$. The set of bases $$\mathcal B$$ satisfies the symmetric exchange property by R. A. Brualdi [Bull. Aust. Math. Soc. 1, 161–167 (1969; Zbl 0172.30703)], namely, if $$B_1$$ and $$B_2$$ are two bases and $$e\in B_1 \setminus B_2$$, then there exists $$f\in B_2 \setminus B_1$$ such that $$B_1 \setminus e \cup f$$ and $$B_2 \setminus f \cup e$$ are bases. The notion of matroid generalizes the notion of independent sets in linear algebra. In [N. L. White, Linear Algebra Appl. 31, 81–91 (1980; Zbl 0458.05022)] the author conjectured that the relations among the symmetric exchanges generate all relations among bases of the matroid. The conjecture is only known for small classes of matroids.
In the present paper, the authors prove the conjecture for strongly base orderable matroids (Theorem 2), i.e., for any two bases $$B_1$$ and $$B_2$$ there is a bijection $$\pi:B_1 \to B_2$$ satisfying the multiple symmetric exchange property. Moreover, they also show that up to saturation the symmetric exchange relations generate all relations among the bases of matroid (Theorem 3). Finally, the authors discuss other versions of White’s conjecture.
Reviewer: Thanh Vu (Lincoln)

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
##### Keywords:
matroid; White’s conjecture; toric ideal
Full Text:
##### References:
 [1] Blasiak, J., The toric ideal of a graphic matroid is generated by quadrics, Combinatorica, 28, 283-297, (2008) · Zbl 1212.05030 [2] Bonin, J., Basis-exchange properties of sparse paving matroids, Adv. in Appl. Math., 50, 1, 6-15, (2013) · Zbl 1256.05037 [3] Brualdi, R. A., Comments on bases in dependence structures, Bull. Aust. Math. Soc., 1, 161-167, (1969) · Zbl 0172.30703 [4] Conca, A., Linear spaces, transversal polymatroids and ASL domains, J. Algebraic Combin., 25, 25-41, (2007) · Zbl 1108.13022 [5] Cox, D.; Little, J.; Schenck, H., Toric varieties, Grad. Stud. Math., vol. 124, (2011), American Mathematical Society Providence · Zbl 1223.14001 [6] Davies, J.; McDiarmid, C., Disjoint common transversals and exchange structures, J. London Math. Soc., 14, 55-62, (1976) · Zbl 0366.05001 [7] Fulton, W., Introduction to toric varieties, Ann. of Math. Stud., vol. 131, (1993), Princeton University Press Princeton [8] Gelfand, I. M.; Goresky, R. M.; MacPherson, R. D.; Serganova, V. V., Combinatorial geometries, convex polyhedra and Schubert cells, Adv. Math., 63, 301-316, (1987) · Zbl 0622.57014 [9] Herzog, J.; Hibi, T., Discrete polymatroids, J. Algebraic Combin., 16, 239-268, (2002) · Zbl 1012.05046 [10] Kapranov, M. M.; Sturmfels, B.; Zelevinsky, A. V., Chow polytopes and general resultants, Duke Math. J., 67, 1, 189-218, (1992) · Zbl 0780.14027 [11] Kashiwabara, K., The toric ideal of a matroid of rank 3 is generated by quadrics, Electron. J. Combin., 17, (2010), RP28, 12 pp · Zbl 1205.05041 [12] Kung, J., Basis-exchange properties, (White, N., Theory of Matroids, Encyclopedia Math. Appl., vol. 26, (1986), Cambridge University Press Cambridge) · Zbl 0587.05018 [13] Lasoń, M., The coloring game on matroids · Zbl 1355.05171 [14] Lasoń, M.; Lubawski, W., On-line List coloring of matroids · Zbl 1358.05109 [15] Michałek, M., Constructive degree bounds for group-based models, J. Combin. Theory Ser. A, 120, 7, 1672-1694, (2013) · Zbl 1316.14098 [16] Oxley, J. G., Matroid theory, Oxford Sci. Publ., (1992), Oxford University Press Oxford · Zbl 0784.05002 [17] Schrijver, A., Combinatorial optimization, polyhedra and efficiency, (2003), Springer-Verlag New York · Zbl 1041.90001 [18] Schweig, J., Toric ideals of lattice path matroids and polymatroids, J. Pure Appl. Algebra, 215, 2660-2665, (2011) · Zbl 1230.13028 [19] Sturmfels, B., Gröbner bases and convex polytopes, Univ. Lecture Ser., vol. 8, (1995), American Mathematical Society Providence [20] Sturmfels, B., Equations defining toric varieties, Proc. Sympos. Pure Math., 62, 437-449, (1997) · Zbl 0914.14022 [21] White, N., The basis monomial ring of a matroid, Adv. Math., 24, 292-297, (1977) · Zbl 0357.05031 [22] White, N., A unique exchange property for bases, Linear Algebra Appl., 31, 81-91, (1980) · Zbl 0458.05022 [23] Woodall, D. R., An exchange theorem for bases of matroids, J. Combin. Theory Ser. B, 16, 227-228, (1974) · Zbl 0275.05020
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.