The behavior of Stanley depth under polarization

2014/01/17 by Bogdan Ichim, Ichim, Bogdan, Lukas Katthän +3 · 1 citation
Computer Science · Mathematics · #05E40 #16W50 #Advanced Combinatorial Mathematics #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1401.4309

openalex publication_date 2014/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let K be a field, R=K[X₁, ..., X_n] be the polynomial ring and J \subsetneq I two monomial ideals in R. In this paper we show that sdepth I/J - depth I/J = sdepth I^p/J^p-depth I^p/J^p, where sdepth I/J denotes the Stanley depth and I^p denotes the polarization. This solves a conjecture by Herzog and reduces the famous Stanley conjecture (for modules of the form I/J) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form R/I and the well-known combinatorial conjecture that every Cohen-Macaulay simplicial complex is partitionable are equivalent.

Citations

• Stanley decompositions and localization
• An algorithm to compute the Hilbert depth

Cited by

• On the Stanley length of monomial ideals

Related

• Stanley depth of monomial ideals
• Values and bounds of the Stanley depth
• Special Stanley Decompositions
• On the Stanley depth of powers of monomial ideals
• Upper bounds for the Stanley depth