Distinguished representations of non-negative polynomials

Abstract

Let g 1 , … , g r ∈ R [ x 1 , … , x n ] such that the set K = { g 1 ⩾ 0 , … , g r ⩾ 0 } in R n is compact. We study the problem of representing polynomials f with f | K ⩾ 0 in the form f = s 0 + s 1 g 1 + ⋯ + s r g r with sums of squares s i , 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.