Theorem 3 A ternary homogeneous symmetric polynomial of degree three $f(x, y, z)$ can be uniquely represented as
$$f(x, y, z)=a g_{3,1}+b g_{3,2}+c g_{3,3},$$

where, $g_{3,1}=\sum x(x-y)(x-z)$,
$$\begin{array}{l} g_{3,2}=\sum(y+z)(x-y)(x-z) \\ g_{3,3}=x y z \end{array}$$

and when $x, y, z \geqslant 0$,
$$a, b, c \geqslant 0 \Leftrightarrow f(x, y, z) \geqslant 0 \text {. }$$