On algorithms for testing positivity of symmetric polynomial functions

We show that positivity on $\mathbb{R}_+^n$ and on $\mathbb{R}^n$ of real symmetric polynomials of degree at most $p$ in $n\ge2$ variables is solvable by algorithms running in $\mathrm{poly}(n)$ time. For real symmetric quartics, we find explicit discriminants and related Maple algorithms running in $\mathrm{lin}(n)$ time.