An isolated umbilical point of the graph of a homogeneous polynomial (Q1590667)

Consider a homogeneous polynomial \(f\) in two real variables \(x,y\) of degree \(k\geq 3\) whose graph \((x,y,f(x,y))\) in \(\mathbb{R}^3\) has the origin \(o\) as an isolated umbilic point. It is proved, that the index \(l\) of \(o\) belongs to the set \(\{1,0, \dots, 1-k/2\}\) if \(k\) is even or \(\{1/2,-1/2, \dots, 1-k/2\}\) if \(k\) is odd. Conversely, for every number \(l\) of the above sets there exists a surface of the above type, so that \(l\) is the index of the isolated umbilic \(o\).