  The $\mathcal{S}$-cone provides a common framework for cones of polynomials
or exponential sums which establish non-negativity upon the
arithmetic-geometric inequality, in particular for sums of non-negative circuit
polynomials (SONC) or sums of arithmetic-geometric exponentials (SAGE). In this
paper, we study the $\mathcal{S}$-cone and its dual from the viewpoint of
second-order representability. Extending results of Averkov and of Wang and
Magron on the primal SONC cone, we provide explicit generalized second-order
descriptions for rational $\mathcal{S}$-cones and theirs duals.
