It concerns the expression of positive definite rational functions as sums of quotients of squares.
[7] A result of Albrecht Pfister[8] shows that a positive semidefinite form in n variables can be expressed as a sum of 2n squares.
[11] McKenna showed in 1975 that all positive semidefinite polynomials with coefficients in an ordered field are sums of weighted squares of rational functions with positive coefficients only if the field is dense in its real closure in the sense that any interval with endpoints in the real closure contains elements from the original field.
[12] A generalization to the matrix case (matrices with polynomial function entries that are always positive semidefinite can be expressed as sum of squares of symmetric matrices with rational function entries) was given by Gondard, Ribenboim[13] and Procesi, Schacher,[14] with an elementary proof given by Hillar and Nie.
It is an open question what is the smallest number such that any n-variate, non-negative polynomial of degree d can be written as sum of at most
[16] The result of Pfister on the other hand fails in the Hermitian case, that is there is no bound on the number of squares required, see D'Angelo–Lebl.