Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classifying the Fine Polyhedral Spectrum

Sofía Garzón Mora, Christian Haase

TL;DR

The paper advances Fine polyhedral adjunction by proving a complete characterization of the largest values in the Fine $\mathds{Q}$-codegree spectrum and, in dimensions $d=1$ and $d=2$, providing full classifications while establishing a general framework for higher dimensions. It combines structural projections (preserving $\mu^{F}$) with a computationalMILP pipeline to enumerate Fine core normal configurations and realize specific numerators, applied to low-volume polytopes from the Bal18 database. In dimension $3$, it derives a concrete spectral form $\mu^{F}(P) = \tfrac{q}{k\ell}$ with $q$ from a finite set and $\ell,k$ in fixed ranges, illustrating the growing complexity beyond $d=2$. The main theoretical contribution links the Fine spectrum to degree-0 and degree-1 polytopes (exceptional simplices and Lawrence prisms), yielding precise bounds and a clean hierarchy of possible values, and it opens questions about spectral gaps and density in higher dimensions with potential toric-geometric implications.

Abstract

In this paper, we examine an analogue of the recently solved spectrum conjecture by Fujita in the setting of Fine polyhedral adjunction theory. We present computational results for lower-dimensional polytopes, which lead to a complete classification of the highest numbers of this Fine spectrum in any dimension. Moreover, we present a classification of the Fine spectrum in dimensions one, two and (almost) three, while providing a framework for general classification results in any dimension.

Classifying the Fine Polyhedral Spectrum

TL;DR

The paper advances Fine polyhedral adjunction by proving a complete characterization of the largest values in the Fine -codegree spectrum and, in dimensions and , providing full classifications while establishing a general framework for higher dimensions. It combines structural projections (preserving ) with a computationalMILP pipeline to enumerate Fine core normal configurations and realize specific numerators, applied to low-volume polytopes from the Bal18 database. In dimension , it derives a concrete spectral form with from a finite set and in fixed ranges, illustrating the growing complexity beyond . The main theoretical contribution links the Fine spectrum to degree-0 and degree-1 polytopes (exceptional simplices and Lawrence prisms), yielding precise bounds and a clean hierarchy of possible values, and it opens questions about spectral gaps and density in higher dimensions with potential toric-geometric implications.

Abstract

In this paper, we examine an analogue of the recently solved spectrum conjecture by Fujita in the setting of Fine polyhedral adjunction theory. We present computational results for lower-dimensional polytopes, which lead to a complete classification of the highest numbers of this Fine spectrum in any dimension. Moreover, we present a classification of the Fine spectrum in dimensions one, two and (almost) three, while providing a framework for general classification results in any dimension.
Paper Structure (15 sections, 15 theorems, 52 equations, 1 table)

This paper contains 15 sections, 15 theorems, 52 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finiteness of the Fine $\mathds{Q}$-codegree spectrum
  4. Tools
  5. Project and Lift
  6. Computational approaches
  7. Results from polytope database
  8. A family of MILPs
  9. Reducing the search space
  10. Realizing a Fine $\mathds{Q}$-codegree numerator
  11. Lattice Pyramids
  12. Classification results
  13. $d=2$
  14. $d=3$
  15. Classification results in dimension $\boldsymbol{d}$

Key Result

Theorem 2.4

mora2023fine Let $P$ be a rational polytope of dimension $d$. The valid relevant inequalities for $P$ of the form $\langle a, \cdot \rangle \geq h^F(a)$ are among the $a \in (\mathbb{Z}^d)^*$ such that $a \in \operatorname{conv}( a_1,...,a_m)$ where the $a_i$ for $1 \leq i \leq m$ are the primitive

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: Paffenholz, Paf15
  • Theorem 2.9
  • Definition 2.10
  • ...and 34 more