Computation of dominant ideals

TL;DR

The paper addresses the problem of when a monomial ideal is dominant and how this property exactly characterizes when the Taylor resolution is minimal. It develops two complementary approaches: combinatorial enumeration under a fixed $\mathrm{lcm}$ and probabilistic sampling via Erdős–Rényi-type models to study dominance in high-dimensional settings. It delivers explicit counting formulas for dominant ideals in low-variable cases, details algorithmic implementations (Macaulay2 and CoCoA), and demonstrates that random sampling can reveal how often dominance occurs depending on the number of generators, degrees, and dimension. This work provides a quantitative, reproducible framework for assessing non-minimality of Taylor resolutions and guides exploration of large spaces of monomial ideals through both exact and stochastic methods.

Abstract

We consider the problem of determining whether a monomial ideal is dominant. This property is critical for determining for which monomial ideals the Taylor resolution is minimal. We first analyze dominant ideals with a fixed least common multiple of generators using combinatorial methods. Then, we adopt a probabilistic approach via the \er\ type model, examining both homogeneous and non-homogeneous cases. This model offers an efficient alternative to exhaustive enumeration, allowing the study of dominance through small random samples, even in high-dimensional settings.

Figures (8)

• Figure 1: Taylor complex (left) and Scarf complex (right) of $I=(x_1x_2, x_2x_3, x_1x_3)$. The Taylor resolution of $I\subset S=\mathbb{K}[x_1,x_2,x_3]$ being $S^1 \xleftarrow[]{\left(x_1^2x_1x_3x_2x_3\right)}S^3 \xleftarrow[]{\left(-x_3-x_2x_30x_10-x_20x_1^2x_1\right)}S^3\xleftarrow[]{\left(x_2-1x_1\right)}S^1.$
• Figure 2:
• Figure 3:
• Figure 4: Frequency of dominant random monomial ideals in $\mathcal{I}{I}(n,D,p)$ for $n=3$, $D=3,\dots,15$ and $p$ taking values in $\{ (D^{-i}-D^{i-1}) /2, \,i=2,3\}\cup \{1/2,1/3,\dots,1/9 \}$. Each data point in the figure represents a sample of size $50$ for a fixed value of $p$.
• Figure 5: Frequency of dominant random homogeneous monomial ideals in $\mathcal{I}_{\mathrm{Gr}}(n,d,p)$ for $n=3$, $d=3,\dots,12$ and the nonzero probabilities taking values in $\left\{ \frac{d^\ell-d^{\ell-1}}{2},\, \ell=2,\dots,n \right\} \cup \left\{ \frac{1}{9},\frac{1}{8},\dots,1 \right\} \cup \left\{\frac{1}{20d},\dots,\frac{20}{20d} \right\}$. Each data point in the figure represents a sample of size $100$ for a fixed value of $p=(0,\dots,0,\alpha)$. The horizontal axes are values of $\alpha$.

Theorems & Definitions (30)

• Definition 1.1: Dominance
• Remark 1.2
• Remark 1.3
• Example 1.4: M2 and CoCoA-5 implementation
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Example 2.3
• Example 2.4