A polynomial $p \in \mathbb{R}[x_1,\dots,x_n]$ is said to be *positive* on a subset $S$ of $\mathbb{R}^n$ if $p(x) > 0$ for every $x \in S$. The polynomial $p$ is called *nonnegative* if $p(x) \ge 0$ for every $x \in S$.

It is known that if $p \in \mathbb{R}[x]$, then $p$ is nonnegative on $\mathbb{R}$ if and only if $p = q^2 + r^2$, with $q,r \in \mathbb{R}[x]$.

I am wondering what is known about the set $$ \mathscr{P} := \{ p \in \mathbb{R}[x]\mid p(x) \ge 0,~\forall x\ge 0 \}; $$ since $f(x) = x \in \mathscr{P}$, it is clear that so this set is distinct from the globally-nonnegative case above.

Any insight/references would be greatly appreciated.