Distinguished representations of non-negative polynomials

Let g1,…,gr∈R[x1,…,xn] such that the set K={g1⩾0,…,gr⩾0} in Rn is compact. We study the problem of representing polynomials f with f|K⩾0 in the form f=s0+s1g1+⋯+srgr with sums of squares si, with particular emphasis on the case where f has zeros in K. Assuming that the quadratic module of all such sums is archimedean, we establish a local-global condition for f to have such a representation, vis-à-vis the zero set of f in K. This criterion is most useful when f has only finitely many zeros in K. We present a number of concrete situations where this result can be applied. As another application we solve an open problem from [S. Kuhlmann et al., Positivity, sums of squares and the multi-dimensional moment problem II, Adv. Geometry, in press] on one-dimensional quadratic modules.