A degenerate isolated fixed point with nonzero local index

The indices. The fixed point equation on $\mathbb{C}$ is $z^2=0$, so $0$ is the only finite fixed point and it is isolated; in the chart $w=1\mathrm{/}z$ at infinity, the fixed point equation is $w^2\mathrm{/}(w+1)=w$, i.e. $w=0$, so $\mathrm{\infty }$ is the other fixed point. By [F1] the two indices are $2$ and $+1$, so the geometric Lefschetz number is $I(f)=2+1=3$; the point $0$ is degenerate, so the determinant formula The index of a nondegenerate fixed point is the sign of det(I-Df) does not apply to it, and the value $2$ comes from the explicit degree computation of the displacement $-z^2$ on a small circle.

The Lefschetz number agrees. The expression $w\mapsto w^2\mathrm{/}(w+1)$ is smooth near infinity, so the polynomial extends smoothly. The finite value $1$ has exactly two distinct preimages solving $z^2+z=1$, namely $(-1\pm \sqrt{5})\mathrm{/}2$; at each the derivative is nonzero complex multiplication by $1+2z$, of positive real determinant. Infinity maps to infinity and is not a preimage of $1$. Hence $1$ is a regular value and Regular-value formula for degree gives degree $2$, so by [F2] $L(f)=1+2=3$. Hence $I(f)=3=L(f)$ even though the fixed point at $0$ is degenerate: the index formula holds for isolated fixed points and does not require nondegeneracy, which is exactly the content of Lefschetz-Hopf index formula.