Local degrees of arbitrary integer value (${\mathrm{deg}}_0(z^k)=k$)

In dimension two, $z\mapsto z^k$ for positive $k$ has local degree $k$ at zero, and $z\mapsto {\overline{z}}^{|k|}$ has degree $k$ for negative $k$. The proper map $z\mapsto |z{|}^2\in \mathbb{C}$ has isolated zero fiber and local degree zero, since its image misses every nearby nonreal target. These proper maps extend to the two-sphere. Taking their product with the identity of ${\mathbb{R}}^{n-2}$ realizes the same local degrees in every dimension $n\ge 2$. In dimension one an isolated local degree is restricted to $0,1,-1$.