1.06 Fundamental theorem of algebra: reprise

(1.20) Define the loop \(\gamma_R(t)=p(Re^{i2\pi t})\). If \(p\) has no complex root then \(\gamma_R\) is a loop in \(\mathbf{C}\setminus\{0\}\). We will prove that the degree \(n\) of \(p\) is equal to zero.

(2.40) As \(R\) varies, this gives a free homotopy. The loop \(\gamma_0\) is the constant loop at \(p(0)\).

(3.20) Since \(\pi_1(\mathbf{C}\setminus\{0\},x\) is abelian, free and based homotopy agree, so we get \([\gamma_0]=[\gamma_R]\in\pi_1(\mathbf{C}\setminus\{0\},x)=\mathbf{Z}\). Since \(\gamma_0\) is constant, \([\gamma_0]=0\).

(4.50) We will show that, for large \(R\), \([\gamma_R]=n\in\mathbf{Z}=\pi_1(\mathbf{C}\setminus\{0\})\). This will imply \[n=0.\] The loop \(\delta_n(t):=R^ne^{i2\pi nt}\) satisfies \([\delta_n]=n\in\mathbf{Z}\), so we need to show \(\gamma_R\simeq\delta_n\) for large \(R\).

(6.24) To achieve this, write \(p(z)=z^n+q(z)\) (\(q(z)=a_{n-1}z^{n-1}+\cdots+a_0\)) and try the homotopy \(H(s,t)=R^ne^{i2\pi nt}+sq(Re^{i2\pi t})\), which connects \(\delta_n\) at \(s=0\) to \(\gamma_R\) at \(s=1\). We need this to be a homotopy in \(\mathbf{C}\setminus\{0\}\), so we need to show that \(|H(s,t)|>0\) for all \(s,t\) when \(R\gg 0\).

(9.00) We estimate:

\begin{align*} |H(s,t)|&\geq ||R^ne^{i2\pi nt}|-s|q(Re^{i2\pi t})||\\ &\geq |R^n-R^{n-1}\max_{k=0}^{n-1}|a_k||, \end{align*}

(10.30) using

\begin{align*} |q(Re^{i2\pi t})|&=|a_{n-1}R^{n-1}e^{2\pi i(n-1)t}+\cdots+a_0|\\ &\leq |a_{n-1}|R^{n-1}+\cdots+|a_0|\\ &\leq R^{n-1}\max_{k=0}^{n-1}(|a_k|). \end{align*}

(12.08) so

\begin{align*} |H(s,t)|&\geq R^{n-1}(R-\max_{k=0}^{n-1}|a_k|), \end{align*}

which is strictly positive whenever \(R\) is strictly greater than \(\max_{k=0}^{n-1}(|a_k|)\).

(13.00) This implies that \(H(s,t)\) is a homotopy in \(\mathbf{C}\setminus\{0\}\) between \(\gamma_0\) (with \([\gamma_0]=0\)) and \(\gamma_R\) (with \([\gamma_R]=n\)), so \(n=0\) and \(p\) is constant, which completes the proof.