Umbilical Points of the Graphs of Homogeneous Polynomials of Degree 3

J. Math. Sci. Univ. Tokyo
Vol. 8 (2001), No. 1, Page 71--87.

Ando, Naoya
Umbilical Points of the Graphs of Homogeneous Polynomials of Degree 3
Let $P^3_o$ be the set of the homogeneous polynomials of degree $3$ such that on their graphs, the origin $o:=(0, 0, 0)$ of {\bf R}$^3$ is isolated as an umbilical point, and $P^{3,1/2}_o , P^{3,-1/2}_o$ the sets of the elements of $P^3_o$ such that on their graphs, the index of $o$ is equal to $1/2, -1/2$, respectively. In this paper, it is seen that $P^3_o$ is divided into $P^{3,1/2}_o$ and $P^{3,-1/2}_o$ by the cone obtained from a rectangular torus in the vector space of the homogeneous polynomials of degree $3$.