Linear spaces, transversal polymatroids and ASL domains

Abstract

We study a class of algebras associated with linear spaces and its relations with polymatroids and integral posets, i.e. posets supporting homogeneous ASL. We prove that the base ring of a transversal polymatroid is Koszul and describe a new class of integral posets. As a corollary we obtain that every Veronese subring of a polynomial ring is an ASL

Universal Gröbner Bases and Cartwright–Sturmfels Ideals

“…In [6] Cartwright and Sturmfels proved that every Z m -graded ideal J with Hilbert series equal to that of I 2 is radical. Their argument relies on the fact that gin(I 2 ) is radical, a result proved by Conca in [8]. This circle of ideas lead us to consider the family of multigraded ideals with a radical gin, and to name them after Cartwright and Sturmfels.…”

“…For I 2 (A) in the row or column graded case, one applies the same approach as above and uses that I 2 (X) ∈ CS(R) for a matrix of variables, a result proved in [8].…”

“…Our results might be seen as the generalization of the Bernstein-Sturmfels-Zelevinsky Theorem [4,14] (asserting that maximal minors of a matrix of variables form a universal Gröbner basis) and of the result of Sturmfels [15] and Villarreal [16] (asserting that the cycles of the complete bipartite graph give rise to the universal Gröbner basis of the ideal of 2-minors of the matrix of variables). Intermediate results in this direction have been obtained in [1,5,6,8,9,12].…”

On Minkowski Sums of Simplices

“…We will subsequently refer to this rhombus as P (2). Now let A be a subset of [r] so that N F (i 1 ) = N F (i 2 ) whenever i 1 and i 2 are in A.…”

Symmetric and quasi-symmetric functions associated to polymatroids

“…6 There exists a homomorphism τ : QSym → Sym [q, t] of commutative algebras such that τ (G[X]) = H[X] for every polymatroid X.…”