Moment Problem and Complexity

Given polynomials h1, . . . , hr ∈ R[x1, . . . , xn], we study the complexity of the problem of representing polynomials in the form f = s0 + s1h1 + · · ·+ srhr (∗) with sums of squares si. Let M be the cone of all f which admit such a representation. The problem is said to be stable if there exists a function φ : N → N such that every f ∈ M has a representation (∗) with deg(si) ≤ φ(deg(f)). The main result says that if the set K = {h1 ≥ 0, . . . , hr ≥ 0} has dimension ≥ 2 and the sequence h1, . . . , hr has the moment property (MP), then the problem is not stable. In particular, this includes the case when K is compact, dim(K) ≥ 2 and the cone M is multiplicatively closed.