A unique exchange property for bases

Abstract

Linear Algebra and Its Applications 31 (1980) 81-91. doi:10.1016/0024-3795(80)90209-8Received by publisher: 1979-07-02Harvest Date: 2016-01-04 12:20:29DOI: 10.1016/0024-3795(80)90209-8Page Range: 81-9

Linear spaces, transversal polymatroids and ASL domains

“…The algebra K[B] is known to be normal and hence CohenMacaulay [11]. White predicted in [26] the shape of the defining equations of K[B] as a quotient of a polynomial ring: they should the quadrics arising from the so-called symmetric exchange relations of the polymatroids. Herzog and Hibi [11] did not "escape from the temptation" to ask whether K[B] is defined by a Gröbner basis of quadrics and whether K[B] is a Koszul algebra.…”

Low degree equations for phylogenetic group-based models

“…Here, also the general question whether the associated ideals are generated by quadrics remains open for over 30 years [Whi80]. However, a recent paper [LM14] by M. Lasoń and the third author proves this result up to saturation, thus providing a description of the corresponding projective toric schemes.…”

On a base exchange game on bispanning graphs

“…Now let us return to our game. White [14] defines the following matroid analogue of the graph game described above: On a block matroid with base pair (X 1 , X 2 ), the maker chooses a ∈ X 1 and the breaker must choose b ∈ X 2 such that X 1 − a + b and X 2 − b + a are new bases for the next move. If after a finite series of moves X 1 and X 2 are exchanged, the maker wins.…”

“…Several authors [3,10,14] analyze the connectivity of some of these graphs. We will describe the four most important types of such graphs and resume results and conjectures on this connectivity problem in the following.…”

“…This game is motivated by questions about the connectivity of several matroid base exchange graphs in [14]. It seems to be surprisingly difficult to study the structure of these graphs already in the graphic case [1,6].…”