The following lemma is what we need to prove the convergence of our bisection approach. It shows that if the Lipschitz constant blows up, the polynomial itself blows up almost everywhere.

Lemma

Let \(R_1(z), R_2(z), \dots\) be a sequence of polynomials of degree at most \(s\) with real coefficients. Denote the coefficients of \(R_i(z)\) by \(a_{ij}\): \[R_i(z) = \sum_{j=0}^s a_{ij} z^j.\] Further, let \(a_i = (a_{i0},a_{i1},\dots,a_{is})^T\) and let the sequence of coefficient vectors have the property that \[\limsup_{i\to\infty}\|a_i\| = \infty.\] Then \(\limsup_{i\to\infty} |R_i(z)| = \infty\) for all but at most \(s\) points \(z\in\mathbf{C}\).

Proof.

Let \(v(z) = (1,z,z^2,\dots,z^s)^T\). Then \(R_i(z) = v(z)\cdot a_i\) and we have \[\Re(R_i(z)) = \|v(z)\| \|a_i\| \cos(\theta_i(z)),\] where \(\theta_i(z)\) is the angle between \(v(z)\) and \(a_i\).

Suppose that \[\limsup_{i\to\infty}|R_i(z)|<\infty\] for some \(z\in\mathbf{C}\). This imples \[\limsup_{i\to\infty}\Re(R_i(z))<\infty,\] so \[\limsup_{i\to\infty}\|v(z)\| \|a_i\| \cos(\theta_i(z))<\infty,\] which implies \(\lim_{i\to\infty}\cos(\theta_i(z))=0\).

Let \(\{z_0,z_1,\dots,z_s\}\in\mathbf{C}\) be distinct but otherwise arbitrary. The vectors \(\{v(z_0),\dots,v(z_s)\}\) are linearly independent since they form the columns of a Vandermonde matrix. Hence \(\lim_{i\to\infty} \cos(\theta_i(z_k))=0\) cannot hold for all \(k=0,1,\dots,s\).