Sums of Squares Methods Explained: Part I

Let p(~x) ∈ R[~x] be SOS in t real polynomial squares. Then, p(~x) must have even degree. Let deg(p(~x)) = 2k. Then, ∃q1, . . . , qt ∈ R[~x] s.t. deg(qi(~x)) ≤ k and p(~x) = ∑t i=1 q 2 i (~x). A key observation is that we can now exactly characterise the finitely many possible power-products that could occur in each qi(~x). Definition 1.1. Let Λn(d) = {α = 〈α1, . . . , αn〉 ∈ Nn | α1 + . . .+ αn ≤ d}. Then, as deg(qi(~x)) ≤ k (∀ 1 ≤ i ≤ t), we see that the exponent vector α for each monomial occurring in each qi(~x) must be a member of Λn(k).