Consider the polynomial $f(x_1, \cdots, x_n)$, I want to characterize $f$ being nonnegative, i.e., $f\geq 0$. For $n=1$, this is equivalent to saying that $f$ is SOS (sum of square). However, in general this does not hold (obviously SOS is only a sufficient, but not necessary condition).
The question is that if I restrict to probability simplices, i.e., $f(x_1, \cdots, x_n)\geq 0$ for all $x_i\in [0,1]$ and $\sum_{i=1}^n x_i=1$, is there a characterization in terms of SOS?