On the Regularity of Dominant and Almost Complete Intersection Monomial Ideals

Abstract: Let $R = k[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $k$, and let $I$ be a monomial ideal of $R$. If $I$ is an almost complete intersection, then we provide an explicit formula for the Castelnuovo-Mumford regularity of $I$ in terms of the powers of the dominant variables appearing in the regular sequence contained in $G(I)$ of length $|G(I)|-1$, where $G(I)$ is the set of minimal monomial generators of $I$. Furthermore, if $I$ is a dominant ideal or an almost complete intersection ideal, then we show that $\operatorname{reg}(\overline{I}) \leq \operatorname{reg}(I),$ where $\overline{I}$ denotes the integral closure of $I$. This provides a positive answer to the Küronya-Pintye conjecture for these two classes of monomial ideals. In addition, we give some examples to clarify these results.

• The paper establishes explicit formulas for Castelnuovo-Mumford regularity for both dominant and ACI monomial ideals using Scarf and Lyubeznik resolutions.
• It verifies that the regularity of the integral closure never exceeds that of the original ideal, confirming the Kűronya–Pintye conjecture for these classes.
• The study offers computational methods validated by numerical examples, advancing practical tools in combinatorial commutative algebra and algebraic geometry.

On the Regularity of Dominant and Almost Complete Intersection Monomial Ideals

Introduction

This paper addresses the computation and comparison of the Castelnuovo-Mumford regularity for two particular classes of monomial ideals in a standard graded polynomial ring, namely dominant ideals and almost complete intersection (ACI) ideals. The work is driven by the Kűronya-Pintye conjecture, which posits that for any homogeneous ideal $I$ in $R = k[x_1, ..., x_n]$, the regularity of its integral closure $\bar{I}$ satisfies $$\operatorname{reg}(\bar{I}) \leq \operatorname{reg}(I)$$. While the conjecture fails in complete generality for $n \geq 4$, it remains open for specific natural classes of monomial ideals. This paper provides explicit regularity formulas for ACI and dominant ideals, and establishes the conjecture for both classes.

Regularity Formulas for Dominant and ACI Monomial Ideals

The authors establish explicit and computationally accessible formulas for the regularity of ACI monomial ideals. An ideal $I = (u_1, ..., u_{q+1})$ is called an almost complete intersection monomial ideal if $u_1, ..., u_q$ form a regular sequence. The main technical results build upon a systematic use of the Scarf and Lyubeznik resolutions, as dominant and semidominant ideals admit minimal resolutions in terms of these constructions.

For dominant monomial ideals, if $G(I) = \{u_1, ..., u_q\}$, the regularity formula is given by:

$$\operatorname{reg}(I) = \sum_{l=1}^q \deg(u_l) - q + 1.$$

For the general ACI case, two scenarios are distinguished:

1. Dominant ACIs: $I$ is itself dominant, so the dominant formula applies.
2. ACI with a generator dividing an LCM: If there exists $R = k[x_1, ..., x_n]$0 for some $R = k[x_1, ..., x_n]$1, then

$R = k[x_1, ..., x_n]$2

where $R = k[x_1, ..., x_n]$3 is the complete intersection part.

This treatment generalizes earlier formulas and provides direct computational methods for cases not previously handled in the literature.

Regularity of Integral Closures and the Kűronya–Pintye Conjecture

A central contribution is the proof that for dominant and almost complete intersection monomial ideals, the regularity of the integral closure does not exceed that of the original ideal, i.e., the Kűronya–Pintye conjecture holds for these classes. This advances previous results, which were restricted to $R = k[x_1, ..., x_n]$4 or special classes such as complete intersections and stable ideals.

The paper utilizes a nuanced analysis of the Newton polyhedron, the structure of minimal generators, and detailed properties of the minimal free resolution. In particular, the possible bases contributing to the maximal degree in the minimal resolution of $R = k[x_1, ..., x_n]$5 are systematically analyzed and compared to those of $R = k[x_1, ..., x_n]$6 itself, leading to a rigorous proof that

$R = k[x_1, ..., x_n]$7

for any dominant or ACI monomial ideal.

The argument confirms that the structural properties of dominant and ACI ideals—specifically how their generators' exponents relate—restrict the possible increase in regularity upon taking the integral closure. The results are reinforced with explicit examples and cross-checked using Macaulay2.

Numerical Illustration

The paper provides non-trivial examples involving monomial ideals in polynomial rings with up to ten variables, where the regularity is computed both theoretically and computationally. In each case, the explicit formulas derived yield accurate regularity counts, and the computed regularity of the integral closure never exceeds that of the original ideal. For instance, an ACI ideal with $R = k[x_1, ..., x_n]$8 in $R = k[x_1, ..., x_n]$9 is shown to have $\bar{I}$0 with $\bar{I}$1, verifying the conjecture numerically.

Implications and Future Directions

The results supply sharp computational tools for the regularity of broad families of monomial ideals, facilitating further advances in both combinatorial commutative algebra and computational algebraic geometry. The explicit regularity bounds for ACIs are significant for the study of syzygies, Rees algebras, and for bounding complexity measures in algorithms involving monomial ideals. The confirmation of the Kűronya-Pintye conjecture for these classes has further theoretical significance, suggesting a strong structural interplay between minimal free resolutions and Newton polyhedron geometry for monomial ideals.

Open directions include the search for further monomial ideal classes for which the conjecture holds, methods to generalize these results to non-monomial cases, and the investigation of the fine structure of the minimal resolution for integral closures beyond the dominant and ACI cases.

Conclusion

The paper systematically establishes explicit regularity formulas for dominant and almost complete intersection monomial ideals and proves that their integral closures have regularity bounded above by that of the original ideal. This extends the known field of validity for the Kűronya-Pintye conjecture and provides effective tools for both theoretical investigation and computational practice in the study of resolutions of monomial ideals (2606.08009).