All vertices for unconstrained symmetric gauge fields

Recently, the generating system that describes interacting symmetric higher-spin gauge fields at the level of equations of motion was proposed. The interaction vertices it offers are 'off the mass shell' unless constrained by the prescribed factorization condition that properly removes traceful components. In this paper we detail the structure of the unconstrained, i.e., traceful vertices. We derive their manifest form to all orders along with a net of the associated dualities, thus providing the complete higher-spin vertex analysis at the unconstrained level for the bosonic theory in any dimension. These vertices are shown to be the minimal space-time local and have a form of the peculiar integrals over a space of closed polygons, which we scrutinize in the paper. The obtained results directly apply to the holomorphic sector of the four-dimensional theory, where the interaction is on shell, producing the all-order chiral higher-spin vertices.