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Three invariants of geometrically vertex decomposable ideals

Thai Thanh Nguyen, Jenna Rajchgot, Adam Van Tuyl

TL;DR

This work develops a recursive framework for three key invariants of geometrically vertex decomposable ideals: the Castelnuovo–Mumford regularity, the multiplicity, and the $a$-invariant. By exploiting geometric vertex decompositions with respect to a variable $y$, and the associated $C_{y,I}$ and $N_{y,I}$ ideals, it derives nondegenerate and degenerate recurrences that tie the invariants of $I$ to those of the smaller ideals, together with a Hilbert-series relation $H_{R/I}(t)=H_{R/(N_{y,I}+iglracket yigr bracket)}(t)+tH_{R/C_{y,I}}(t)$. The authors show that $a(R/I)\le 0$, hence all such ideals are almost Hilbertian, and recover known results (e.g., regularity for pure vertex-decomposable Stanley–Reisner ideals) as corollaries; they also apply the recursions to toric ideals of bipartite graphs to obtain concrete formulas for regularity, $a$-invariant, and multiplicity in important graph families (Ferrers graphs, gluing cycles, and $G_{r,d}$). The approach provides new, uniform proofs of several prior results and highlights the close interplay between combinatorial decompositions and algebraic invariants.

Abstract

We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the $a$-invariant. We show that these invariants can be computed recursively using the ideals that appear in the geometric vertex decomposition process. As an application, we prove that the $a$-invariant of a geometrically vertex decomposable ideal is non-positive. We also recover some previously known results in the literature including a formula for the regularity of the Stanley--Reisner ideal of a pure vertex decomposable simplicial complex, and proofs that some well-known families of ideals are Hilbertian. Finally, we apply our recursions to the study of toric ideals of bipartite graphs. Included among our results on this topic is a new proof for a known bound on the $a$-invariant of a toric ideal of a bipartite graph.

Three invariants of geometrically vertex decomposable ideals

TL;DR

This work develops a recursive framework for three key invariants of geometrically vertex decomposable ideals: the Castelnuovo–Mumford regularity, the multiplicity, and the -invariant. By exploiting geometric vertex decompositions with respect to a variable , and the associated and ideals, it derives nondegenerate and degenerate recurrences that tie the invariants of to those of the smaller ideals, together with a Hilbert-series relation . The authors show that , hence all such ideals are almost Hilbertian, and recover known results (e.g., regularity for pure vertex-decomposable Stanley–Reisner ideals) as corollaries; they also apply the recursions to toric ideals of bipartite graphs to obtain concrete formulas for regularity, -invariant, and multiplicity in important graph families (Ferrers graphs, gluing cycles, and ). The approach provides new, uniform proofs of several prior results and highlights the close interplay between combinatorial decompositions and algebraic invariants.

Abstract

We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the -invariant. We show that these invariants can be computed recursively using the ideals that appear in the geometric vertex decomposition process. As an application, we prove that the -invariant of a geometrically vertex decomposable ideal is non-positive. We also recover some previously known results in the literature including a formula for the regularity of the Stanley--Reisner ideal of a pure vertex decomposable simplicial complex, and proofs that some well-known families of ideals are Hilbertian. Finally, we apply our recursions to the study of toric ideals of bipartite graphs. Included among our results on this topic is a new proof for a known bound on the -invariant of a toric ideal of a bipartite graph.
Paper Structure (11 sections, 31 theorems, 66 equations, 3 figures)

This paper contains 11 sections, 31 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background on Geometrically Vertex Decomposable Ideals
  3. Regularity of Geometrically Vertex Decomposable Ideals
  4. Multiplicity of Geometrically Vertex Decomosable Ideals
  5. The $a$-invariant and the Hilbertian property
  6. Applications to toric ideals of graphs
  7. Background on toric ideals of graphs
  8. Toric ideals of bipartite graphs
  9. Regularity and gluing cycles
  10. Regularity of toric ideals of Ferrers graphs
  11. Invariants for a class of bipartite graphs

Key Result

Theorem 1.1

Suppose that $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ is a homogeneous, geometrically vertex decomposable ideal and $\text{in}_y(I) = C_{y,I} \cap (N_{y,I}+\langle y \rangle)$ is its geometric vertex decomposition. If the decomposition is non-degenerate, then When the geometric vertex decomposition is degenerate, we have

Figures (3)

  • Figure 1: Two graphs $G$ and $C_4$ glued along edges $e$ and $f$
  • Figure 2: The graph $T_\lambda$ for $\lambda = (3,3,3,2)$
  • Figure 3: Illustration of $G_{r,d}=G_{6,5}$. This graph is a $K_{2,5}$ with a path of length $2\cdot6 -2=10$ connecting the two vertices of degree 5 in $K_{2,5}$.

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2: Corollary \ref{['cor.nonpos']}
  • Theorem 1.3: Theorems \ref{['thm.regbipartite']} and \ref{['thm.toricbipartitehilbertan']}
  • Theorem 1.4: Theorem \ref{['thm.Ferrer']}
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 71 more