Non-existence of degree bounds for weighted sums of squares representations

Given a fixed family of polynomials h(1),..., h(r) epsilon R vertical bar x(1),...,x(n)}, we study the problem of representing polynomials in the form. f = s(0) s(1)h(1) + (...) + s(r)h(r) 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 phi N -> N Such that every f epsilon M has a representation (*) with de-(s(i)) <= phi(deg(f)). The main result says that if the, subset K = {h(1) >= 0,..., h(r) >= 0} of R-n has dimension >= 2 and the sequence h(1),..., h(r). has the moment property (MP), then the problem is not stable. In particular, this includes the case where K is compact, dim(K) >= 2 and the cone M is multiplicatively closed. (c) 2005 Elsevier Inc. All rights reserved.