Cyclic polytope of the simplest cubic fields

TL;DR

This paper investigates the lattice point enumeration within dilated cyclic polytopes derived from generators of the simplest cubic fields, providing exact counts for specific dilation ranges.

Contribution

It introduces a novel approach to counting lattice points in cyclic polytopes associated with simplest cubic fields, with explicit formulas for certain dilations.

Findings

Exact lattice point counts for specific dilations

Identification of dilation ranges with precise lattice point enumeration

Insights into the structure of cyclic polytopes in algebraic number fields

Abstract

In this paper, we study dilation of cyclic polytopes with the vertices defined by a generator of the simplest cubic fields. In particular, for a specific range of values, we give a precise number of the contained lattice points.