On the gradient of a monomial ideal

2025/11/19 by Antonino Ficarra, Ficarra, Antonino
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2511.15879

openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28

Abstract

Let K be a field of characteristic zero, let I ⊂ S = K[x₁,…,x_n] be a homogeneous ideal, and let ∂(I) be its gradient ideal. We study the relationship between reg I and reg ∂(I). While earlier work by Busé, Dimca, Schenck, and Sticlaru showed these regularities are generally incomparable for hypersurface ideals, we prove they remain incomparable even for monomial ideals with linear resolution, answering a question of J. Herzog. In fact, for any integers a ∈ ℤ and b ≥ - 1, we construct monomial ideals I and J such that reg I - reg ∂(I) = a, reg ∂(J) - reg J = b and J has linear resolution. We introduce monomial ideals with differential linear resolution as those monomial ideals whose all iterated gradient ideals have linear resolution. We prove that polymatroidal ideals, equigenerated (strongly) stable ideals, powers of edge ideals with linear resolution, complementary edge ideals with linear resolution, and certain equigenerated squarefree monomial ideals with many generators satisfy this property.

Citations

• Rees algebras of complementary edge ideals
• Stanley-Reisner ideals with linear powers
• Complementary edge ideals
• Componentwise linear symbolic powers of edge ideals and Minh's conjecture
• A new proof of the Herzog-Hibi-Zheng theorem
• The canonical trace of Cohen-Macaulay algebras of codimension 2
• Shellability of Componentwise Discrete Polymatroids
• The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in ℙⁿ
• Symbolic powers of cover ideal of very well-covered and bipartite graphs
• The Golod property for powers of ideals and Koszul ideals
• Ordinary and symbolic powers are Golod
• Intersections of Leray complexes and regularity of monomial ideals
• A generalization of the Taylor complex construction
• Stanley--Reisner rings with large multiplicities are Cohen--Macaulay
• Monomial ideals whose powers have a linear resolution

Related

• A note on linear resolution and polymatroidal ideals
• Monomial ideals whose powers have a linear resolution
• Bounded powers of edge ideals: regularity and linear quotients
• The regularity of monomial ideals and their integral closures
• Generalized mixed product ideals whose powers have a linear resolution