  The moment-SOS (sum of squares) hierarchy is a powerful approach for solving
globally non-convex polynomial optimization problems (POPs) at the price of
solving a family of convex semidefinite optimization problems (called
moment-SOS relaxations) of increasing size, controlled by an integer, the
relaxation order. We say that a relaxation of a given order is exact if solving
the relaxation actually solves the POP globally. In this note, we study the
geometry of the exactness cone, defined as the set of polynomial objective
functions for which the relaxation is exact. Generalizing previous foundational
work on quadratic optimization on real varieties, we prove by elementary
arguments that the exactness cones are unions of semidefinite representable
cones monotonically embedded for increasing relaxation order.
