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Unimodality of the $h^*$-vector for unimodular triangulations whose boundary is an induced subcomplex

Mykola Pochekai

TL;DR

This work addresses the unimodality of the Ehrhart $h^*$-vector for unimodular lattice triangulations with boundary induced as a subcomplex. It develops a top-heavy Lefschetz framework in a generic Artinian reduction of the face ring, supported by an exact unreduced partition complex restricted to interior vertices, to establish first-half monotonicity of the $h^*$-vector and, combined with boundary-induced exactness, full unimodality. The proofs leverage a Čech-type partition complex, strong Lefschetz elements on stars, and injection properties across graded components to derive the required inequalities. Together these results advance the understanding of unimodality in Ehrhart theory under structural conditions on triangulations, providing a concrete method to obtain unimodality in this class of polytopes.

Abstract

We prove that the Ehrhart $h^*$-vector is unimodal for unimodular triangulations whose boundary is an induced subcomplex.

Unimodality of the $h^*$-vector for unimodular triangulations whose boundary is an induced subcomplex

TL;DR

This work addresses the unimodality of the Ehrhart -vector for unimodular lattice triangulations with boundary induced as a subcomplex. It develops a top-heavy Lefschetz framework in a generic Artinian reduction of the face ring, supported by an exact unreduced partition complex restricted to interior vertices, to establish first-half monotonicity of the -vector and, combined with boundary-induced exactness, full unimodality. The proofs leverage a Čech-type partition complex, strong Lefschetz elements on stars, and injection properties across graded components to derive the required inequalities. Together these results advance the understanding of unimodality in Ehrhart theory under structural conditions on triangulations, providing a concrete method to obtain unimodality in this class of polytopes.

Abstract

We prove that the Ehrhart -vector is unimodal for unimodular triangulations whose boundary is an induced subcomplex.
Paper Structure (2 sections, 9 theorems, 16 equations)

This paper contains 2 sections, 9 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. The Proof

Key Result

Theorem 1.1

Let $B$ be a unimodular lattice $d$-ball triangulation whose boundary is an induced subcomplex. Then there exists a top-heavy Lefschetz element $h$ in a generic Artinian reduction $A^*(B)$ of its face ring $\mathbb {R}^*[B]$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 3 more