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Asymptotic invariants of symbolic powers of binomial edge ideals

Dennis Belotserkovskiy, Mariana Landín, Charlie Ruppe, Lizzy Teryoshin

TL;DR

This work analyzes the asymptotic behavior of symbolic powers for binomial edge ideals and their monomial analogues, focusing on the Waldschmidt constant and asymptotic regularity. By employing the symbolic polyhedron framework, the authors connect these invariants to induced subgraphs and demonstrate that $\widehat{\alpha}(J_G)=\widehat{\alpha}(\text{in}_<(J_G))=2$ for every graph and that $\widehat{\alpha}(\text{gin}(J_G))=\widehat{\alpha}(I_G)$, linking binomial-edge and edge-ideal settings. They provide subgraph-based inequalities for Betti numbers of symbolic powers and establish a unifying route to bound $\widehat{\alpha}$ via clique and chromatic numbers, with explicit results for classes like closed and complete multipartite graphs. The symbolic polyhedron formalism enables modular analysis across induced subgraphs, yielding both theoretical insights and practical bounds relevant to algebraic combinatorics and commutative algebra. Open problems invite exploration of asymptotic regularity phenomena beyond the currently understood two-parameter regime and conjecture tight relationships with induced-path lengths.

Abstract

To a graph $G$ one associates the binomial edge ideal $J_G$ generated by a collection of binomials corresponding to the edges of $G$. In this paper, we study the asymptotic behavior of symbolic powers of $J_G$, its lexicographic initial ideal $\mathrm{in}_<(J_G)$, and its multigraded generic initial ideal $\mathrm{gin}(J_G)$. We focus on the Waldschmidt constant, $\widehatα$, and asymptotic regularity, $\widehat{\mathrm{reg}}$, which capture linear growth of minimal generator degrees and Castelnuovo--Mumford regularity. We explicitly compute $\widehatα(J_G)$ and $\widehatα(\mathrm{in}_<(J_G))$, and compare the Betti numbers of the symbolic powers of $J_G$ and $J_H$, where $H$ is a subgraph of $G$. To analyze $\mathrm{in}_<(J_G)$ and $\mathrm{gin}(J_G)$, we use the symbolic polyhedron, a convex polyhedron that encodes the elements of the symbolic powers of a monomial ideal. We determine its vertices via $G$'s induced connected subgraphs and show that $\widehatα(\mathrm{gin}(J_G))=\widehatα(I_G)$, where $I_G$ is the edge ideal of $G$. This yields an alternate proof of known bounds for $\widehatα(I_G)$ in terms of $G$'s clique number and chromatic number.

Asymptotic invariants of symbolic powers of binomial edge ideals

TL;DR

This work analyzes the asymptotic behavior of symbolic powers for binomial edge ideals and their monomial analogues, focusing on the Waldschmidt constant and asymptotic regularity. By employing the symbolic polyhedron framework, the authors connect these invariants to induced subgraphs and demonstrate that for every graph and that , linking binomial-edge and edge-ideal settings. They provide subgraph-based inequalities for Betti numbers of symbolic powers and establish a unifying route to bound via clique and chromatic numbers, with explicit results for classes like closed and complete multipartite graphs. The symbolic polyhedron formalism enables modular analysis across induced subgraphs, yielding both theoretical insights and practical bounds relevant to algebraic combinatorics and commutative algebra. Open problems invite exploration of asymptotic regularity phenomena beyond the currently understood two-parameter regime and conjecture tight relationships with induced-path lengths.

Abstract

To a graph one associates the binomial edge ideal generated by a collection of binomials corresponding to the edges of . In this paper, we study the asymptotic behavior of symbolic powers of , its lexicographic initial ideal , and its multigraded generic initial ideal . We focus on the Waldschmidt constant, , and asymptotic regularity, , which capture linear growth of minimal generator degrees and Castelnuovo--Mumford regularity. We explicitly compute and , and compare the Betti numbers of the symbolic powers of and , where is a subgraph of . To analyze and , we use the symbolic polyhedron, a convex polyhedron that encodes the elements of the symbolic powers of a monomial ideal. We determine its vertices via 's induced connected subgraphs and show that , where is the edge ideal of . This yields an alternate proof of known bounds for in terms of 's clique number and chromatic number.
Paper Structure (11 sections, 36 theorems, 86 equations, 2 figures)

This paper contains 11 sections, 36 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Binomial edge ideals
  4. Symbolic powers and asymptotic invariants
  5. Symbolic polyhedra
  6. Results for binomial edge ideals and the initial ideal
  7. Symbolic polyhedra
  8. Vertices of the symbolic polyhedron
  9. Decomposing the symbolic polyhedra
  10. The Waldschmidt constant of the multigraded generic initial ideal
  11. Open Problems

Key Result

Theorem 1

For all graphs $G$, $\widehat{\alpha}(J_G) = \widehat{\alpha}(\mathrm{in}(J_G)) = 2$. For closed graphs $G$ with lexicographic ordering $<$, $\widehat{\mathop{\mathrm{reg}}\nolimits}(J_G)=\widehat{\mathop{\mathrm{reg}}\nolimits}(\mathop{\mathrm{in_<}}\nolimits(J_G))=2$.

Figures (2)

  • Figure 1: Symbolic polyhedra corresponding to the ideals $(xy,xz,yz)$, $(x,y,z)$, and $(xz,y)$ respectively.
  • Figure 2: The net graph and an induced subgraph

Theorems & Definitions (78)

  • Theorem 1: \ref{['thm: waldschmidt J_G']}, \ref{['closed_graph_cor']}
  • Theorem 2: \ref{['cor:subgraph_decomp']}
  • Theorem 3: \ref{['thm:chi_omega_bound']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: herzog2010binomial
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.7
  • Theorem 2.8: herzog2010binomial, conca
  • ...and 68 more