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

摘要

Given a fixed family of polynomials h_1, …, h_r ∈R [x_1, …, x_n], we study the problem of representing polynomials in the form f = s_0 + s_1h_1 + … + s_rh_r with sums of squares s_i. 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(s_i) ≤ φ (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.