  In this work we study positivity preservers
$T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ with constant
coefficients and define their generators $A$ if they exist, i.e., $\exp(A) =
T$. We use the theory of regular Fr\'echet Lie groups to show the first main
result. A positivity preserver with constant coefficients has a generator if
and only if it is represented by an infinitely divisible measure (Main Theorem
4.7). In the second main result (Main Theorem 4.11) we use the L\'evy--Khinchin
formula to fully characterize the generators of positivity preservers with
constant coefficients.
