Degree of map
Posted: Tue Nov 10, 2015 1:11 pm
Let \(\displaystyle{p}\) be a polynomial function on \(\displaystyle{\mathbb{C}}\) which has no root on \(\displaystyle{S^{1}}\) .
Show that the number of roots of the equation \(\displaystyle{p(z)=0\,,z\in\mathbb{C}}\) with \(\displaystyle{|z|<1}\)
is the degree of the map \(\displaystyle{\bar{p}:S^{1}\longrightarrow S^{1}}\) specified by \(\displaystyle{\bar{p}(z)=\dfrac{p(z)}{|p(z)|}}\) .
Show that the number of roots of the equation \(\displaystyle{p(z)=0\,,z\in\mathbb{C}}\) with \(\displaystyle{|z|<1}\)
is the degree of the map \(\displaystyle{\bar{p}:S^{1}\longrightarrow S^{1}}\) specified by \(\displaystyle{\bar{p}(z)=\dfrac{p(z)}{|p(z)|}}\) .