A criterion of containment for tropical hypersurfaces

Abstract

For tropical $n$-variable polynomials $f,g$ a criterion of containment for tropical hypersurfaces $Trop(f)\subset Trop(g)$ is provided in terms of their Newton polyhedra $N(f),N(g)\subset {\mathbb{R}}^{n+1}$. Namely, $Trop(f)\subset Trop(g)$ iff for every vertex $v$ of $N(g)$ there exist a homothety $t\cdot N(f),t>0$ and a parallel shift $s:{\mathbb{R}}^{n+1}\to {\mathbb{R}}^{n+1}$ such that $v\in s(t\cdot N(f))\subset N(g)$.