  Positivity preservation is an important issue in the dynamics of open quantum
systems: positivity violations always mark the border of validity of the model.
We investigate the positivity of self-adjoint polynomial Gaussian integral
operators $\widehat{\kappa}_{PG}$, that is, the multivariable kernel
$\kappa_{PG}$ is a product of a polynomial $P$ and a Gaussian kernel
$\kappa_G$. These operators frequently appear in open quantum systems.
  We show that $\widehat{\kappa}_{PG}$ can be only positive if the Gaussian
part is positive, which yields a strong and quite easy test for positivity.
This has an important corollary for the bipartite entanglement of the density
operators $\widehat{\kappa}_{PG}$: if the Gaussian density operator
$\widehat{\kappa}_G$ fails the Peres-Horodecki criterion, then the
corresponding polynomial Gaussian density operators $\widehat{\kappa}_{PG}$
also fail the criterion for all $P$, hence they are all entangled.
  We prove that polynomial Gaussian operators with polynomials of odd degree
cannot be positive semidefinite.
  We introduce a new preorder $\preceq$ on Gaussian kernels such that if
$\kappa_{G_0}\preceq \kappa_{G_1}$ then $\widehat{\kappa}_{PG_0}\geq 0$ implies
$\widehat{\kappa}_{PG_1}\geq 0$ for all polynomials $P$. Therefore, deciding
the positivity of a polynomial Gaussian operator determines the positivity of a
lot of another polynomial Gaussian operators having the same polynomial factor,
which might improve any given positivity test by carrying it out on a much
larger set of operators. We will show an example that this really can make
positivity tests much more sensitive and efficient. This preorder has
implication for the entanglement problem, too.
