[Paper Review] Values and bounds of the Stanley depth

Muhammad Ishaq|arXiv (Cornell University)|Oct 22, 2010

Commutative Algebra and Its Applications12 references3 citations

TL;DR

This paper establishes new upper and lower bounds for the Stanley depth of monomial ideals in polynomial rings, proving that the Stanley depth of an ideal is bounded above by the minimum Stanley depth of its associated prime ideals. It confirms the Stanley conjecture in cases where associated primes are generated by disjoint variable sets, even for non-squarefree ideals, and provides tight bounds via combinatorial decomposition techniques and recursive reduction using colon ideals.

ABSTRACT

We give different bounds for the Stanley depth of a monomial ideal $I$ of a polynomial algebra $S$ over a field $K$. For example we show that the Stanley depth of $I$ is less or equal with the Stanley depth of any prime ideal associated to $S/I$. Also we show that the Stanley conjecture holds for $I$ and $S/I$ when the associated prime ideals of $S/I$ are generated by disjoint sets of variables.

Motivation & Objective

• To establish tight upper and lower bounds for the Stanley depth of monomial ideals.
• To investigate conditions under which the Stanley conjecture holds, particularly when associated prime ideals are generated by disjoint sets of variables.
• To extend known results on Stanley depth to non-squarefree monomial ideals.
• To provide computable bounds using colon ideals and recursive decomposition techniques.

Proposed method

• Uses the colon ideal technique: for a monomial $ w \notin I $, $ \operatorname{sdepth}(I) \leq \operatorname{sdepth}(I:w) $, and applies this to associated primes of $ S/I $.
• Applies known results on Stanley depth of prime ideals, such as $ \operatorname{sdepth}(P) = n - \lfloor \operatorname{ht}(P)/2 \rfloor $, to bound $ \operatorname{sdepth}(I) $.
• Employs recursive reduction via $ I:v $ for suitable monomials $ v $, reducing the problem to lower-dimensional subrings.
• Uses the decomposition $ \operatorname{sdepth}(I) \leq \operatorname{sdepth}(I' \cap K[x_1,\dots,x_r]) + (n - r) $ to bound depth in higher dimensions.
• Applies combinatorial bounds from [7] and [6] to derive expressions involving binomial coefficients and floor functions for multi-ideal intersections.
• Generalizes results to intersections of four or more prime ideals with disjoint generators, deriving a complex but tight upper bound.

Experimental results

Research questions

• RQ1 Under what conditions does the Stanley conjecture hold for a monomial ideal $ I $ and its quotient $ S/I $?
• RQ2 How can the Stanley depth of $ I $ be bounded in terms of the Stanley depths of its associated prime ideals?
• RQ3 What are effective upper bounds for $ \operatorname{sdepth}(I) $ when $ I $ is an intersection of three or more prime ideals with non-disjoint generators?
• RQ4 Can tight bounds be derived for $ \operatorname{sdepth}(I) $ when the associated primes are not generated by disjoint variables?
• RQ5 What is the behavior of $ \operatorname{sdepth}(I) $ when $ I $ is generated by $ m $ monomials and $ \operatorname{ht}(P) = m $ for some associated prime $ P $?

Key findings

• The Stanley depth of a monomial ideal $ I $ is bounded above by the minimum Stanley depth of its associated prime ideals: $ \operatorname{sdepth}(I) \leq \min_{1 \leq i \leq s} \operatorname{sdepth}(P_i) $.
• When $ I $ has $ m $ minimal generators and $ \operatorname{ht}(P) = m $ for some associated prime $ P $, then $ \operatorname{sdepth}_S(I) = n - \lfloor m/2 \rfloor $.
• If the minimal generators of the associated primes of $ S/I $ are pairwise disjoint, the Stanley conjecture holds for both $ I $ and $ S/I $, even when $ I $ is not squarefree.
• For three associated primes $ P_1, P_2, P_3 $ with disjoint generators, the paper provides a tight upper bound $ \operatorname{sdepth}(I) \leq d $, where $ d $ is a complex expression involving $ n, t, r, s, q $, and floor functions.
• In the case of four pairwise disjoint prime ideals, the paper derives an upper bound involving binomial coefficients: $ \operatorname{sdepth}(I) \leq 3 + d + \frac{1}{d_1 d_2 d_3} \left[ \binom{n-d}{4} - \sum \binom{d_i}{4} - \sum \binom{d_i}{3}(n - d_i) - \sum \binom{d_i}{2}\binom{d_j}{2} \right] $.
• In Example 3.12, the bounds yield $ \operatorname{sdepth}(I) \in [10, 11] $, and using [7, Corollary 2.2], it is concluded that $ \operatorname{sdepth}(I) = 11 $.