  We investigate the problem of representing moment sequences by measures in
the context ofPolynomial Optimization Problems. This consists in finding the
infimum of a real polynomial ona real semialgebraic set defined by polynomial
inequalities. We analyze the exactness of MomentMatrix (MoM) relaxations, dual
to the Sum of Squares (SoS) relaxations, which are hierarchiesof convex cones
introduced by Lasserre to approximate measures and positive polynomials.
Weinvestigate in particular flat truncation properties, which allow testing
effectively when MoMexactness holds.We consider the quadratic module Q
generated by the inequalities. We show that the dualof the MoM relaxation
coincides with the SoS relaxation extended with the real radical ofthe support
of Q, and focus on the zero-dimensional case, generalizing results for
equationsdefining a finite real variety. We deduce sufficient and necessary
conditions for flat truncation,under the finite convergence assumption: flat
truncation happens if and only if the support ofthe quadratic module associated
with the minimizers is of dimension zero. We also bound theorder of the
relaxation at which flat truncation holds.As corollaries, we conclude that flat
truncation holds:$\bullet$ when regularity conditions, known as Boundary
Hessian Conditions, hold: this resultimplies that flat truncation and MoM
exactness holds generically;$\bullet$ when the support of the quadratic module
Q is zero-dimensional;$\bullet$ in singular cases, flat truncation holds for
the MoM relaxation extended with the polarconstraints when the real variety of
polar points is finite.Effective numerical computations illustrate these flat
truncation properties.
