On the Polymatroidal Property of Monomial Ideals with a View Towards Orderings of Minimal Generators

Abstract

We prove that a monomial ideal I generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also conjecture that the polymatroidal ideals can be characterized with linear quotients property with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables. We prove our conjecture in many special cases.2010 Mathematics Subject Classification. 13F20; 05E40.

Homological shifts of polymatroidal ideals

“…It is known that a polymatroidal ideal I ⊂ S has linear quotients with respect to the lexicographic order induced by [3,Theorem 2.4]. Using the description of HS 1 (I) given in Proposition 1.7 we can prove the following result.…”

“…Then I has linear quotients with respect to > lex , [3,Theorem 2.4]. For u ∈ G(I), we denote by set(u) the following set {i :…”