Sums of Squares: Methods for Proving Identity Families

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...

Sums of distinct squares

Sums-of-Squares Formulas

The following is the extended version of my notes from my ATC talk given on June 4, 2014 at UCLA. I begin with a basic introduction to sums-of-squares formulas, and move on to giving motivation for studying these formulas and discussing some results about them over the reals. More recent techniques have made it possible to obtain similar results over arbitrary fields, and some of these are disc...

Sums of Two Squares

n = 1: 1 = 0 + 1; n = 2 (prime): 2 = 1 + 1; n = 3 (prime) is not a sum of two squares. n = 4: 4 = 2 + 0. n = 5 (prime): 5 = 2 + 1. n = 6 is not a sum of two squares. n = 7 (prime) is not a sum of two squares. n = 8: 8 = 2 + 2. n = 9: 9 = 3 + 0. n = 10: 10 = 3 + 1. n = 11 (prime) is not a sum of two squares. n = 12 is not a sum of two squares. n = 13 (prime): 13 = 3 + 2. n = 14 is not a sum of t...

Refined Asymptotics for Sparse Sums of Squares

To prove that a polynomial is nonnegative on R one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than classical algebraic methods (see, e.g., [Par03]), thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact...