  We establish a simple criterion for locating points where the transition
density of a degenerate diffusion is strictly positive. Throughout, we assume
that the diffusion satisfies a stochastic differential equation (SDE) on
$\mathbf{R}^d$ with additive noise and polynomial drift. In this setting, we
will see that it is often that case that local information of the flow, e.g.
the Lie algebra generated by the vector fields defining the SDE at a point
$x\in \mathbf{R}^d$, determines where the transition density is strictly
positive. This is surprising in that positivity is a more global property of
the diffusion. This work primarily builds on and combines the ideas of Ben
Arous and L\'eandre (1991) and Jurdjevic and Kupka (1981, 1985).
