A criterion of containment for tropical hypersurfaces

For tropical n-variable polynomials $f, g$ a criterion of containment for tropical hypersurfaces Trop(f)⊂ Trop(g) is provided in terms of their Newton polyhedra N(f), N(g)⊂ Rⁿ⁺¹. Namely, Trop(f)⊂ Trop(g) iff for every vertex v of $N(g)$ there exist a homothety t· N(f), t>0 and a parallel shift s:Rⁿ⁺¹ → Rⁿ⁺¹ such that v∈ s(t· N(f))⊂ N(g).