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Failure of the Lefschetz property for the Graphic Matroid

Ryo Takahashi

TL;DR

The paper discredits the Maeno–Numata conjecture in the graphic-matroid setting by constructing counterexamples where the basis generating polynomial $f_G$ of a graph yields a degenerate higher Hessian, hence failing the strong Lefschetz property at degree $i=3$. It combines probabilistic screening via Schwartz–Zippel with a constructive deterministic verification that produces an explicit kernel vector $m F$ for $m{H}_{B_3}(f_G)$, certifying degeneracy for a graph with $n=13$ variables and socle degree $d=7$ (Hilbert function $(1,13,70,166,166,70,13,1)$). The paper also demonstrates partial failure of $ ext{SLP}_2$ for a specific eight-vertex graph under a fixed Lefschetz element $oldsymbol{ extell}=oldsymbol{1}$, illustrating the nuanced behavior of Lefschetz properties in graphic cases. Overall, these results reveal limitations of the conjecture in graphic matroids and showcase a computational approach that ties Hessian degeneracy to explicit kernel constructions. The findings sharpen understanding of when Lefschetz properties hold and highlight the complexity of extending matroid-based Lefschetz predictions.

Abstract

We consider the strong Lefschetz property for standard graded Artinian Gorenstein algebras. Such an algebra has a presentation of the quotient algebra of the ring of the differential polynomials modulo the annihilator of some homogeneous polynomial. There is a characterization of the strong Lefschetz property for such an algebra by the non-degeneracy of the higher Hessian matrix of the homogeneous polynomial. Maeno and Numata conjectured that if such an algebra is defined by the basis generating polynomial of any matroid, then it has the strong Lefschetz property. For this conjecture, we give counterexamples that are associated with graphic matroids. We prove the degeneracy of the higher Hessian matrix by constructing a non-zero element in the kernel of that matrix.

Failure of the Lefschetz property for the Graphic Matroid

TL;DR

The paper discredits the Maeno–Numata conjecture in the graphic-matroid setting by constructing counterexamples where the basis generating polynomial of a graph yields a degenerate higher Hessian, hence failing the strong Lefschetz property at degree . It combines probabilistic screening via Schwartz–Zippel with a constructive deterministic verification that produces an explicit kernel vector for , certifying degeneracy for a graph with variables and socle degree (Hilbert function ). The paper also demonstrates partial failure of for a specific eight-vertex graph under a fixed Lefschetz element , illustrating the nuanced behavior of Lefschetz properties in graphic cases. Overall, these results reveal limitations of the conjecture in graphic matroids and showcase a computational approach that ties Hessian degeneracy to explicit kernel constructions. The findings sharpen understanding of when Lefschetz properties hold and highlight the complexity of extending matroid-based Lefschetz predictions.

Abstract

We consider the strong Lefschetz property for standard graded Artinian Gorenstein algebras. Such an algebra has a presentation of the quotient algebra of the ring of the differential polynomials modulo the annihilator of some homogeneous polynomial. There is a characterization of the strong Lefschetz property for such an algebra by the non-degeneracy of the higher Hessian matrix of the homogeneous polynomial. Maeno and Numata conjectured that if such an algebra is defined by the basis generating polynomial of any matroid, then it has the strong Lefschetz property. For this conjecture, we give counterexamples that are associated with graphic matroids. We prove the degeneracy of the higher Hessian matrix by constructing a non-zero element in the kernel of that matrix.
Paper Structure (11 sections, 3 theorems, 21 equations, 6 figures, 1 table)

This paper contains 11 sections, 3 theorems, 21 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strong Lefschetz Property
  4. Basis Generating Polynomial
  5. Failure of the $\mathrm{SLP}_3$
  6. Screening
  7. Verification
  8. Construction Algorithm of $\bm F$
  9. The table of $\bm F$
  10. The list of Counterexample Candidates
  11. Partial Failure of the $\mathrm{SLP}_2$

Key Result

Theorem 2.4

Let $k\le d/2$ be a non-negative integer and $B_k=\{\alpha_1,\dots,\alpha_m\}$ be any $\mathbb{R}$-basis of $A_k$. For any $(a_1,\dots,a_n)\in\mathbb{R}^n$, the algebra $A$ has the $\mathrm{SLP}_k$ with an element $\ell=a_1\partial_1+\dots+a_n\partial_n\in A_1$ if and only if $\bm{H}_{B_k}(f)(a_1,\d

Figures (6)

  • Figure 1: A graph corresponding to one of the counterexamples
  • Figure 2: The graph with the smallest number of edges without $\mathrm{SLP}_3$
  • Figure 3: A planar graph without $\mathrm{SLP}_3$
  • Figure 4: The candidate graph with the largest number of edges
  • Figure 5: The graph in \ref{['figure:maximal candidate']} contains the graph in \ref{['figure:SLP_3']}
  • ...and 1 more figures

Theorems & Definitions (14)

  • Definition 2.1: strong Lefschetz property (in the narrow sense)
  • Remark 2.2
  • Definition 2.3: higher Hessian matrix
  • Theorem 2.4: harima2013lefschetz*Theorem 3.76,MR2594646*Theorem 3.1,watanabe2000hessian*Theorem 4
  • proof
  • Remark 2.5
  • Example 2.6: ikeda1996results*Example 4.4
  • Definition 2.7: basis generating polynomial (for graphs)
  • Conjecture 2.8: Maeno--Numata conjecture (for graphs) maeno2012sperner*conjecture
  • Remark 2.9
  • ...and 4 more