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Thin Simplices via Modular Arithmetic

Vadym Kurylenko

TL;DR

The paper addresses the classification of lattice simplices with vanishing local $h^*$-polynomials (thin simplices) by translating the problem into modular arithmetic and linear coding theory. It develops a correspondence between simplices and extended linear codes over $ obreak\mathbb{Z}_N$ and recasts thinness in terms of the absence of maximal-weight codewords and the emptiness of a hyperplane complement, enabling a code-theoretic route to classification. Leveraging this framework, the authors achieve a complete description of 4-dimensional thin simplices not expressible as free joins or pyramids, revealing six sporadic instances plus a one-parameter width-1 family, and proving the existence of infinitely many such 4D examples. The work highlights how width, spanning properties, and Cayley structure interact with thinness, and it raises open questions about finiteness and structure of thin polytopes in higher dimensions.

Abstract

The local $h^*$-polynomial is a natural invariant of a lattice polytope appearing in Ehrhart theory and Hodge theory. In this work, we study the question posed in [GKZ94] concerning the classification of lattice simplices with vanishing local $h^*$-polynomial. Such simplices are called thin. We relate this question to linear codes and hyperplane arrangements over finite rings. This allows us to obtain a complete classification of the $4$-dimensional thin simplices, extending the previously known results in dimensions up to $3$.

Thin Simplices via Modular Arithmetic

TL;DR

The paper addresses the classification of lattice simplices with vanishing local -polynomials (thin simplices) by translating the problem into modular arithmetic and linear coding theory. It develops a correspondence between simplices and extended linear codes over and recasts thinness in terms of the absence of maximal-weight codewords and the emptiness of a hyperplane complement, enabling a code-theoretic route to classification. Leveraging this framework, the authors achieve a complete description of 4-dimensional thin simplices not expressible as free joins or pyramids, revealing six sporadic instances plus a one-parameter width-1 family, and proving the existence of infinitely many such 4D examples. The work highlights how width, spanning properties, and Cayley structure interact with thinness, and it raises open questions about finiteness and structure of thin polytopes in higher dimensions.

Abstract

The local -polynomial is a natural invariant of a lattice polytope appearing in Ehrhart theory and Hodge theory. In this work, we study the question posed in [GKZ94] concerning the classification of lattice simplices with vanishing local -polynomial. Such simplices are called thin. We relate this question to linear codes and hyperplane arrangements over finite rings. This allows us to obtain a complete classification of the -dimensional thin simplices, extending the previously known results in dimensions up to .
Paper Structure (19 sections, 27 theorems, 130 equations, 5 tables)

This paper contains 19 sections, 27 theorems, 130 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Local $h^*$-polynomial
  3. Thin simplices
  4. Main result
  5. Experiments and questions
  6. Structure of the paper
  7. Lattice simplices and modular arithmetic
  8. Our Setup
  9. Linear codes
  10. From simplices to linear codes
  11. From simplices to hyperplane arrangements
  12. Spanning thin simplices
  13. Lattice width
  14. Four-dimensional thin simplices
  15. General strategy of classification
  16. ...and 4 more sections

Key Result

Theorem 1.3

Let $\Delta$ be a four-dimensional thin simplex that is not a free join. Then $\Delta$ is either one of the $6$ sporadic cases in the Table sporadictable or it belongs to the one-parameter family of non-spanning simplices of width $1$ given below the table.

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem 1.3
  • Conjecture 1.5
  • Conjecture 1.6
  • Example 1.7
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: batyrev_lattice_2013, Theorem 2.3.
  • Example 2.5
  • Definition 2.6
  • ...and 47 more