On bifurcations of homogeneous polinomial vector fields on the plane

The paper considers vector fields on the plane whose components are homogeneous polynomials of degree $n$. The set $HP_n$ of such vector fields is identified with the $2 n + 2-$dimensional space of coefficients of these polynomials. Phase portraits of vector fields are viewed on the projective plane. Structurally stable vector fields $X$ , for which the topological structure of the phase portrait does not change when passing to a vector field close enough to $X$ in $HP_n$, form an open everywhere dense set in $HP_n$. In this paper, we describe an open everywhere dense set in the subspace of structurally unstable vector fields in $HP_n$. It is an analytic submanifold of codimension one in $HP_n$ and consists of vector fields $X$ of the first degree of structural instability, for which the topological structure of the phase portrait does not change when passing to a structural unstable vector field close enough to $X$ in $HP_n$. We describe bifurcations for vector fields of the first degree structurally instability.