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Using complex coordinates near zero, $F_k(z)=z^k$ for $k>0$ induces winding number $k$ on a small circle, so its local relative homology multiplier is $k$. For $k<0$, $F_k(z)={\overline{z}}^{|k|}$ has local degree $k$, since conjugation reverses the winding. For zero, take $F_0(z)=|z{|}^2\in \mathbb{C}$: its zero fiber is the single point zero, while the image of a small circle is a single positive real point, so its local degree is zero. Equivalently a nearby nonreal target has no preimages.

Each of these maps is proper and hence extends continuously over the one-point compactifications to a map $S^2\to S^2$ with zero as an isolated preimage of zero. ThusThese are local degrees of arbitrary integer value. Taking $F_k$ times the identity on ${\mathbb{R}}^{d-2}$ gives the same local degrees for every $d\ge 2$. In dimension one, an isolated local degree is restricted to $-1,0,1$, so the all-integers assertion is realized in dimensions at least two.