Algebraic certificates of (semi)definiteness for polynomials over fields containing the rationals

Sum of squares (SOS) decompositions for positive semidefinite polynomials are usually computed numerically, using convex optimization solvers. The precision of the decompositions can be improved by increasing the number of digits used in the computations, but, when the number of variables is greater than the length (i.e., the minimum number of squares needed for the decomposition) of the polynomial, it is difficult to obtain an exact SOS decomposition with the existing methods. A new algorithm, which works well in "almost all" such cases, is proposed here. The results of randomly generated experiments are reported to compare the proposed algorithm with those based on convex optimization.