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Two lower bounds for the Stanley depth of monomial ideals
Author(s) -
Katthän L.,
Seyed Fakhari S. A.
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400269
Subject(s) - mathematics , conjecture , monomial , monomial ideal , combinatorics , vertex (graph theory) , dimension (graph theory) , lattice (music) , simplicial complex , order (exchange) , ideal (ethics) , discrete mathematics , polynomial ring , polynomial , graph , mathematical analysis , philosophy , physics , finance , epistemology , acoustics , economics
Let J ⊈ I be two monomial ideals of the polynomial ring S = K [ x 1 , ... , x n ] . In this paper, we provide two lower bounds for the Stanley depth of I / J . On the one hand, we introduce the notion of lcm number of I / J , denoted by l ( I / J ) , and prove that the inequality sdepth ( I / J ) ≥ n − l ( I / J ) + 1 holds. On the other hand, we show that sdepth ( I / J ) ≥ n − dim L I / J, where dim L I / Jdenotes the order dimension of the lcm lattice of I / J . We show that I and S / I satisfy Stanley's conjecture, if either the lcm number of I or the order dimension of the lcm lattice of I is small enough. Among other results, we also prove that the Stanley–Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley's conjecture.