Vertex-Maximal Lattice Polytopes Contained in 2-Simplices

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.

13 pages, 7 figures