Testing containment of tropical hypersurfaces within polynomial complexity

Abstract

For tropical n-variable polynomials $f,g$ a criterion of containment for tropical hypersurfaces $\mathrm{Trop}\left(f\right)\subset \mathrm{Trop}\left(g\right)$ is provided in terms of their Newton polyhedra $N\left(f\right),N\left(g\right)\subset R^{n+1}$. Namely, $\mathrm{Trop}\left(f\right)\subset \mathrm{Trop}\left(g\right)$ iff for every vertex v of $N\left(g\right)$ there exists a unique vertex w of $N\left(f\right)$ such that for the tangent cones it holds $v-w+N{\left(f\right)}_w\subseteq N{\left(g\right)}_v$. Relying on this criterion an algorithm is designed which tests whether $\mathrm{Trop}\left(f\right)\subset \mathrm{Trop}\left(g\right)$ within polynomial complexity.