Necessary and sucient conditions for symmetric homogeneous polynomial inequalities of degree four and six in real variables

Let fn(x;y;z) be a symmetric homogeneous polynomial of degree n=4 or n=6, in three real variables. We give necessary and sufficient conditions to have fn(x;y;z)≥0 for all real numbers x;y;z. Then, we apply the obtained results to prove several relevant symmetric homogeneous polynomial inequalities of degree four and six.