  We study $K$-positivity preservers with given closed
$K\subseteq\mathbb{R}^n$, i.e., linear maps
$T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that
$T\mathrm{Pos}(K)\subseteq\mathrm{Pos}(K)$ holds, and their generators
$A:\mathrm{R}[x_1,\dots,x_n]\to\mathrm{R}[x_1,\dots,x_n]$, i.e.,
$e^{tA}\mathrm{Pos}(K)\subseteq\mathrm{Pos}(K)$ holds for all $t\geq 0$. We
characterize these maps $T$ for any closed $K\subseteq\mathbb{R}^n$ in Theorem
4.5. We characterize the maps $A$ in Theorem 5.8 for $K=\mathbb{R}^n$ and give
partial results for general $K$. In Example 5.10 we give a map $A$ such that
$e^{tA}$ is a positivity preserver for all $t\geq \tau$ for some $\tau>0$ but
not for $t\in (0,\tau)$, i.e., we have an eventually positive semi-group.
