Two-parameter unfolding of a parabolic point of a vector field in $\mathbb C$ fixing the origin

Abstract: In this paper we describe the bifurcation diagram of the$2$-parameter family of vector fields $\dot z = z(z^k+\epsilon_1z+\epsilon_0)$ over $\mathbb C\mathbb P^1$ for $(\epsilon_1,\epsilon_0)\in \mathbb C^2$. There are two kinds of bifurcations: bifurcations of parabolic points and bifurcations of homoclinic loops through infinity. The latter are studied using the tool of the periodgon introduced in a particular case in \cite{CR}, and then generalized in \cite{KR}. We apply the results to the bifurcation diagram of a generic germ of 2-parameter analytic unfolding preserving the origin of the vector field $\dot z = z^{k+1} +o(z^{k+1})$ with a parabolic point at the origin.