An intrinsic characterization of the Bogdanov-Takens normal-form coefficients and a mixed-volume obstruction to non-isolated degeneracies

Víctor Castellanos, Ramón Eduardo Chan-López

Let $X$ be a planar vector field with an equilibrium $p$ at which the Jacobian $J=DX(p)$ is nilpotent of rank one, and let $q_0$ span $\ker J$. We prove that the two coefficients $…