Completions, reversals, and duality for tropical varieties

The object of this paper is to present two algebraic results with straightforward proofs, which have interesting consequences in tropical geometry. We start with an identity for polynomials over the max-plus algebra, which shows that any polynomial divides a product of binomials. Interpreted in tropical geometry, any tropical variety W can be completed to a union of tropical primitives, i.e. single-face polyhedral complexes. In certain situations, a tropical variety W has a "reversal" variety, which together with W already yields the union of primitives; this phenomenon is explained in terms of a map defined on the algebraic structure, and yields a duality on tropical hypersurfaces.