**Abstract:** We consider a polynomial vector field F in the plane and begin by showing that the trajectories of F may be piecewise trivialized. In fact this result holds in far greater generality than polynomial F, namely we need only assume that F is definable in an o-minimal structure to derive the trivialization. Given this we may abstract the situation for such a vector field F by attaching to it a set of first order logical axioms TF that describe the geometric behaviour of the trajectories. Under appropriate finiteness conditions for the closed trajectories of F we show that TF is a set of axioms of finite rank, where rank is interpreted in the recently developed context of thorn forking. We also have a converse, namely TF being of the appropriate rank implies the finiteness conditions for closed trajectories of F alluded to above. Finally we point out how the logical considerations relevant to the theory TF lead to numerous questions in the general model theory of densely ordered structures. This is joint work with Patrick Speissegger. Host: C.W. Henson