Ideals with linear quotients and componentwise polymatroidal ideals
Somayeh Bandari, Ayesha Asloob Qureshi
TL;DR
This paper investigates the relationship between ideals with linear quotients and componentwise polymatroidal ideals, providing classes where the converse implication holds, thus advancing understanding in monomial ideal theory.
Contribution
The paper identifies two classes of ideals—componentwise polymatroidal ideals in two variables and those with strong exchange property—where the converse of a known implication is true.
Findings
In $K[x,y]$, componentwise polymatroidal ideals have linear quotients.
Componentwise polymatroidal ideals with strong exchange property also have linear quotients.
The paper expands the classes of ideals for which the converse implication holds.
Abstract
If is a monomial ideal with linear quotients, then it has componentwise linear quotients. However, the converse of this statement is an open question. In this paper, we provide two classes of ideals for which the converse of this statement holds. First class is the componentwise polymatroidal ideals in and the second one is the componentwise polymatroidal ideals with strong exchange property.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Rings, Modules, and Algebras · Polynomial and algebraic computation