Topological degree of homogeneous function of degree k [closed]

Question

Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k? This is true if F is polynomial!

Answer 1

No. Consider the map $F(r\,\cos(\theta),r\,\sin(\theta))=(r\cos(n\theta),r\,\sin(n\theta))$. This is homogeneous of degree $1$ but has topological degree $n$.