SOS-convexity

A multivariate polynomial is SOS-convex (or sum of squares convex) if its Hessian matrix H can be factored as H("x") = "S"T("x")"S"("x") where "S" is a matrix (possibly rectangular) which entries are polynomials in "x". In other words, the Hessian matrix is a SOS matrix polynomial.
An equivalent definition is that the form defined as "g"("x","y") = "y"TH("x")"y" is a sum of squares of forms.
Connection with convexity.
If a polynomial is SOS-convex, then it is also convex. Since establishing whether a polynomial is SOS-convex amounts to solving a semidefinite programming problem, SOS-convexity can be used as a proxy to establishing if a polynomial is convex. In contrast, deciding if a generic polynomial of degree large than four is convex is a NP-hard problem.
The first counterexample of a polynomial which is convex but not SOS-convex was constructed by Amir Ali Ahmadi and Pablo Parrilo in 2009. The polynomial is a homogeneous polynomial that is sum-of-squares and given by:formula_1In the same year, Grigoriy Blekherman proved in a non-constructive manner that there exist convex forms that is not representable as sum of squares. An explicit example of a convex form (with degree 4 and 272 variables) that is not a sum of squares was claimed by James Saunderson in 2021.
Connection with non-negativity and sum-of-squares.
In 2013 Amir Ali Ahmadi and Pablo Parrilo showed that every convex homogeneous polynomial in "n" variables and degree 2"d" is SOS-convex if and only if either (a) "n" = 2 or (b) 2"d" = 2 or (c) "n" = 3 and 2"d" = 4. Impressively, the same relation is valid for non-negative homogeneous polynomial in "n" variables and degree 2"d" that can be represented as sum of squares polynomials (See Hilbert's seventeenth problem).