[Paper Review] Regularity and h-polynomials of binomial edge ideals

Takayuki Hibi, Kazunori Matsuda|arXiv (Cornell University)|Aug 21, 2018

Commutative Algebra and Its Applications10 references7 citations

TL;DR

This paper constructs finite simple graphs whose binomial edge ideals exhibit arbitrary prescribed Castelnuovo–Mumford regularity and h-polynomial degree, proving that for any integers $ r $ and $ s $ with $ 2 \leq r \leq s $, there exists a graph $ G $ such that $ \operatorname{reg}(K[\mathbf{x},\mathbf{y}]/J_G) = r $ and $ \deg h_{K[\mathbf{x},\mathbf{y}]/J_G}(\lambda) = s $. The construction relies on iterated suspensions and starlike graphs to control these invariants independently.

ABSTRACT

Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, \ldots, n \}$ and $K[X, Y] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\mathrm{deg} x_i = \mathrm{deg} y_j = 1$. The binomial edge ideal of $G$ is the binomial ideal $J_G \subset K[X, Y]$ which is generated by those binomials $x_iy_j - x_jy_i$ for which $\{i, j\}$ is an edge of $G$. The Hilbert series $H_{K[X, Y]/J_G}(λ)$ of $K[X, Y]/J_G$ is of the form $H_{K[X, Y]/J_G}(λ) = h_{K[X, Y]/J_G}(λ)/(1 - λ)^d$, where $d = \mathrm{dim} K[X, Y]/J_G$ and where $h_{K[X, Y]/J_G}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$ with each $h_i \in \mathbb{Z}$ and with $h_s eq 0$ is the $h$-polynomial of $K[X, Y]/J_G$. It is known that, when $K[X, Y]/J_G$ is Cohen-Macaulay, one has $\mathrm{reg}(K[X, Y]/J_G) = \mathrm{deg} h_{K[X, Y]/J_G}(λ)$, where $ \mathrm{reg}(K[X, Y]/J_G)$ is the (Castelnuovo-Mumford) regularity of $K[X, Y]/J_G$. In the present paper, given arbitrary integers $r$ and $s$ with $2 \leq r \leq s$, a finite simple graph $G$ for which $\mathrm{reg}(K[X, Y]/J_G) = r$ and $\mathrm{deg} h_{K[X, Y]/J_G}(λ) = s$ will be constructed.

Motivation & Objective

• To resolve the open problem of realizing arbitrary pairs of regularity and h-polynomial degree for binomial edge ideals.
• To extend previous results on monomial ideals to the more complex setting of binomial edge ideals.
• To provide explicit graph-theoretic constructions that control algebraic invariants like regularity and h-polynomials.
• To demonstrate that the regularity and h-polynomial degree can be independently prescribed for binomial edge ideals of finite simple graphs.

Proposed method

• Using the suspension operation on graphs to incrementally increase the degree of the h-polynomial while preserving regularity.
• Constructing t-starlike graphs from paths and a central vertex to achieve high h-polynomial degree relative to regularity.
• Applying known formulas for Hilbert series and h-polynomials of binomial edge ideals of complete graphs and path graphs.
• Leveraging results from [7], [10], and [11] on regularity and Hilbert series of edge ideals under join and suspension operations.
• Using induction on the number of suspended vertices to prove that regularity remains bounded while h-polynomial degree increases.
• Analyzing the height of minimal primes in the binomial edge ideal to show non-Cohen–Macaulayness in certain constructions, yet preserving the desired invariants.

Experimental results

Research questions

• RQ1 Can every pair of integers $ r, s $ with $ 2 \leq r \leq s $ be realized as the regularity and h-polynomial degree of a binomial edge ideal?
• RQ2 What graph constructions allow independent control over the regularity and h-polynomial degree of $ K[\mathbf{x},\mathbf{y}]/J_G $?
• RQ3 How does the suspension operation affect the regularity and h-polynomial of binomial edge ideals?
• RQ4 Is it possible to construct non-Cohen–Macaulay binomial edge ideals with prescribed regularity and h-polynomial degree?
• RQ5 What role do starlike graphs play in achieving a gap between regularity and h-polynomial degree?

Key findings

• For any integers $ r $ and $ s $ with $ 2 \leq r \leq s $, there exists a finite simple graph $ G $ such that $ \operatorname{reg}(K[\mathbf{x},\mathbf{y}]/J_G) = r $ and $ \deg h_{K[\mathbf{x},\mathbf{y}]/J_G}(\lambda) = s $, proving the main theorem.
• The t-starlike graph $ T_{p_1,\dots,p_t} $ satisfies $ \operatorname{reg}(K[\mathbf{x},\mathbf{y}]/J_G) = 2 + \sum p_i $ and $ \deg h_{K[\mathbf{x},\mathbf{y}]/J_G}(\lambda) = t + \sum p_i $, yielding a strict inequality between regularity and h-polynomial degree when $ t \geq 3 $.
• Iterated suspension of a path graph preserves regularity but increases the h-polynomial degree by one at each step, enabling fine control over the degree.
• The graph $ G_m $ on $ m+9 $ vertices is not unmixed (hence not Cohen–Macaulay), yet satisfies $ \operatorname{reg}(K[\mathbf{x},\mathbf{y}]/J_{G_m}) = \deg h_{K[\mathbf{x},\mathbf{y}]/J_{G_m}}(\lambda) = m+6 $, showing the invariants can be equal even in non-Cohen–Macaulay cases.
• The construction of $ G $ via iterated suspension and starlike graphs ensures that $ \operatorname{reg}(K[\mathbf{x},\mathbf{y}]/J_G) \leq \deg h_{K[\mathbf{x},\mathbf{y}]/J_G}(\lambda) $, with strict inequality possible.
• The paper establishes that the regularity and h-polynomial degree are not linked by the Cohen–Macaulay property in general, as demonstrated by non-Cohen–Macaulay examples with equal invariants.