Denote by the space of real polynomials in of degree Sufficient conditions, linear in the coefficients of polynomial ( ) are provided in order that be a sum of squares (sos) of polynomials.
Write where is a constant, the i-th standard vector, and free of the monomials Let be the even lattice points with Theorem: If the conditions i) and ii) hold true, then is sos.
To prove this and similar further results the following equivalence is noted:
of degree is sos whenever the moment matrix defined by ( the reals) is positive semidefinite.
Then is inferred from i, ii and elementary but technical estimates for given These latter are partially in J. B. Lasserre and T. Netzer [“Sos approximation of nonnegative polynomials via simple high degree perturbations”, Math. Z. 256, 99–112 (2006; Zbl 1122.13005)]. Since the author saves space at some wrong places, the reader may have to spend some time to fill in crucial (though short) arguments.
V. Powers and T. Wörmann [“An algorithm for sums of squares of real polynomials”, J. Pure Appl. Algebra 127, 99–104 (1998; Zbl 0936.11023)] (based on ideas of Choi-Lam-Reznick) have given a necessary and sufficient criterion for a polynomial to be sum of squares and proposed Tarski quantifier elimination (q.e.) for deciding the criterion. Parrilo saw that it is in fact a semidefinite programming problem if the are numerically given. However, q.e. would creep in again if the coefficients are treated as indeterminates and likely lead to very complicated conditions. Therefore results like the theorem above are interesting.