Two lower bounds for the Stanley depth of monomial ideals

Autor(en): Katthaen, L.
Fakhari, S. A. Seyed
Stichwörter: DECOMPOSITIONS; lcm lattice; lcm number; Mathematics; Monomial ideal; order dimension; simplicial complex; Stanley depth
Erscheinungsdatum: 2015
Herausgeber: WILEY-V C H VERLAG GMBH
Journal: MATHEMATISCHE NACHRICHTEN
Volumen: 288
Ausgabe: 11-12
Startseite: 1360
Seitenende: 1370
Zusammenfassung: 
Let J?I be two monomial ideals of the polynomial ring S=K[x1,...,xn]. 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-dimLI/J, where dimLI/J denotes 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. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISSN: 0025584X
DOI: 10.1002/mana.201400269

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