An Isolated Umbilical Point of the Graph of a Homogeneous Polynomial

Suppose that the origin o of R ³ is an isolated umbilical point of the graph of a homogeneous polynomial in two real variables of degree k3. Then we see that the index of o is an element of the set 1–k/2+i ^k/2] _i=0. Moreover, we see that each element of 1–k/2+i ^k/2] _i=0 may be the index of o on the graph of a suitable homogeneous polynomial of degree k.