Vertex-Maximal Lattice Polytopes Contained in 2-Simplices

TL;DR

This paper investigates the maximum number of vertices in lattice polytopes contained within scaled 2-simplices, providing exact results for an infinite subset of scaling factors and bounds for others.

Contribution

It introduces a family of vertex-maximal lattice polytopes within scaled 2-simplices and establishes bounds for cases not covered by the family.

Findings

Determined maximum vertices for an infinite subset of scalings

Constructed explicit vertex-maximal polytopes

Provided bounds for remaining cases

Abstract

Motivated by the problem of bounding the number of rays of plane tropical curves we study the following question: Given $n\in\mathbb{N}$ and a unimodular $2$-simplex $\Delta$ what is the maximal number of vertices a lattice polytope contained in $n\cdot \Delta$ can have? We determine this number for an infinite subset of $\mathbb{N}$ by providing a family of vertex-maximal polytopes and give bounds for the other cases.