Determine the winding number of the following map: \(f: S^1 \to \mathbb{C}-\{0\}, z\mapsto 8z^4 + 4z^3 + 2z^2 + z^{-1}\)
Solution
1. On \(S^1\), write\[ f(z)=z^{-1}g(z),\qquad g(z)=8z^5+4z^4+2z^3+1. \]
This is immediate by multiplying the given Laurent polynomial by \(z\).
2. The polynomial \(g\) has all five zeros in the open unit disk.
On \(|z|=1\), \[ |8z^5|=8>4+2+1\ge |4z^4+2z^3+1|. \] By Rouché’s theorem, \(g(z)\) and \(8z^5\) have the same number of zeros in \(|z|<1\), counted with multiplicity, namely five. The strict inequality also shows that \(g\) has no zero on \(S^1\).
3. The winding number of \(g|_{S^1}\) about \(0\) is \(5\).
By the argument principle, the winding number of a polynomial with no zeros on the contour equals the number of its zeros inside the contour, counted with multiplicity. Apply 2.
4. The winding number of \(z\mapsto z^{-1}\) is \(-1\), so\[ \boxed{\operatorname{wind}(f,0)=5-1=4}. \]
Winding number is additive under pointwise multiplication of maps \(S^1\to\mathbb C^*\). The map \(z^{-1}\) has degree \(-1\), while 3 gives degree \(5\) for \(g\).