Maximal integral simplices with no interior integer points

TL;DR

This paper classifies and explores the properties of integral maximal lattice-free simplices, showing a complete classification in three dimensions and finiteness results in higher dimensions, leading to a conjecture about their overall finiteness.

Contribution

It provides a complete classification of 3D integral maximal lattice-free simplices and proves finiteness for a subclass in higher dimensions, proposing a conjecture for the general case.

Findings

In 3D, only seven types of integral maximal lattice-free simplices exist up to unimodular transformations.

The set of such simplices with vertices on coordinate axes is finite in higher dimensions.

Conjecture: the total number of these simplices is finite in any dimension.

Abstract

In this paper, we consider integral maximal lattice-free simplices. Such simplices have integer vertices and contain integer points in the relative interior of each of their facets, but no integer point is allowed in the full interior. In dimension three, we show that any integral maximal lattice-free simplex is equivalent to one of seven simplices up to unimodular transformation. For higher dimensions, we demonstrate that the set of integral maximal lattice-free simplices with vertices lying on the coordinate axes is finite. This gives rise to a conjecture that the total number of integral maximal lattice-free simplices is finite for any dimension.