Reversibility and Quasi-Homogeneous Normal Forms of Vector Fields

This paper uses tools in Quasi-Homogeneous Normal Form theory to discuss certain aspects of reversible vector fields around an equilibrium point. Our main result provides an algorithm, via Lie Triangle, that detects the non-reversibility of vector fields. As a consequence we answer an intriguing question related to the problems derived from the $16^{\circ}$ Hilbert Problem. That is, it is possible to decide whether a planar center is not reversible. Some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields. We find several families of nilpotent centers which are non-reversible.