[Paper Review] How to compute the Stanley depth of a monomial ideal

Jürgen Herzog, Marius Vlădoiu+1 more|ArXiv.org|Dec 14, 2007

Commutative Algebra and Its Applications8 references3 citations

TL;DR

This paper establishes that the Stanley depth of a quotient of monomial ideals $I/J$ in a polynomial ring can be computed in finite time by analyzing partitions of a finite poset derived from the monomials in $I \setminus J$. The key contribution is a constructive method using interval partitions of a characteristic poset to compute both Stanley depth and a related invariant called $\operatorname{fdepth}$, which is defined via prime filtrations.

ABSTRACT

Let $J\subset I$ be monomial ideals. We show that the Stanley depth of $I/J$ can be computed in a finite number of steps. We also introduce the $\fdepth$ of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps. In both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals.

Motivation & Objective

• To establish a finite algorithm for computing the Stanley depth of $I/J$ where $J \subset I$ are monomial ideals.
• To define and compute a new invariant $\operatorname{fdepth}$ for monomial ideals using prime filtrations.
• To show that both Stanley depth and $\operatorname{fdepth}$ can be determined by analyzing interval partitions of a finite poset associated with $I/J$.
• To provide a constructive framework for computing these invariants by reducing them to combinatorial problems on posets.

Proposed method

• Define the characteristic poset $P^{g}_{I/J}$ as the set of integer vectors $a \leq g$ such that $x^a \in I \setminus J$, for a suitable $g \in \mathbb{Z}^n$.
• Show that every partition of $P^{g}_{I/J}$ into intervals induces a Stanley decomposition of $I/J$.
• Prove that for any Stanley decomposition of $I/J$, there exists an induced decomposition from such a partition whose Stanley depth is at least as large.
• Introduce $\operatorname{fdepth}$ as the maximum, over all prime filtrations of $I/J$, of the minimum dimension of the associated prime ideals.
• Establish that $\operatorname{fdepth}(I/J)$ can be computed by restricting to partitions of $P^{g}_{I/J}$ whose partial unions form poset ideals.
• Use Corollary 2.8 to characterize the relevant partitions that yield optimal $\operatorname{fdepth}$ and Stanley depth values.

Experimental results

Research questions

• RQ1 Can the Stanley depth of a quotient $I/J$ of monomial ideals be computed algorithmically in finite time?
• RQ2 Is there a combinatorial characterization of Stanley depth in terms of interval partitions of a finite poset?
• RQ3 Can the $\operatorname{fdepth}$ invariant, defined via prime filtrations, also be computed in finite time?
• RQ4 What is the relationship between Stanley depth and $\operatorname{fdepth}$, and how do they relate to the depth of the module?
• RQ5 Does the conjecture that $\operatorname{sdepth} \mathfrak{m} = \lceil n/2 \rceil$ for the maximal ideal $\mathfrak{m}$ in $n$ variables hold in general?

Key findings

• The Stanley depth of $I/J$ can be computed by examining all interval partitions of the characteristic poset $P^{g}_{I/J}$, and the maximum Stanley depth among such decompositions is the true Stanley depth.
• For any Stanley decomposition of $I/J$, there exists a decomposition induced by a poset partition whose Stanley depth is at least as large, ensuring that the optimal value is captured by this method.
• The $\operatorname{fdepth}$ of $I/J$ can be computed in finite time by analyzing only those interval partitions of $P^{g}_{I/J}$ whose partial unions are poset ideals.
• For the graded maximal ideal $\mathfrak{m} \subset S = K[x_1,\dots,x_n]$, the paper shows $\operatorname{sdepth} \mathfrak{m} = \lceil n/2 \rceil$ for $n \leq 9$, supporting a conjecture on the general case.
• The paper proves that $\operatorname{sdepth}I = n-1$ for any monomial complete intersection ideal $I$ minimally generated by three elements in $n$ variables.
• The authors show that Soleyman Jahan's conjecture on degree bounds in Stanley decompositions implies a new conjecture stating that the minimal exponents in the intervals of a poset partition are bounded by $\operatorname{reg}(I/J)$.