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

Given a fixed family of polynomials h1,…,hr∈R[x1,…,xn], we study the problem of representing polynomials in the form(*)f=s0+s1h1+⋯+srhrwith 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 subset K={h1⩾0,…,hr⩾0} of Rn 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 where K is compact, dim(K)⩾2 and the cone M is multiplicatively closed.