Classifying the Fine Polyhedral Spectrum

TL;DR

This paper advances the understanding of the Fine polyhedral spectrum by providing classifications in low dimensions and computational results that inform the spectrum's behavior across all dimensions.

Contribution

It offers the first complete classification of the highest Fine spectrum numbers in any dimension and develops a framework for future classifications.

Findings

Complete classification of highest spectrum numbers in all dimensions.

Classification of Fine spectrum in dimensions one, two, and nearly three.

Computational results for lower-dimensional polytopes.

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.