= Local degrees of arbitrary integer value
{title2=$\deg_0(z^k)=k$}
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 $n\geq2$. In dimension one an isolated local degree is restricted to $0,1,-1$.
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