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 $v\in s(t\cdot N(f))\subset N(g)$.