\section{The data of pair \texorpdfstring{\ref{pair:K2N18}}{I}}\label{app:pairK2N18}

The zeros are
\[
  b=-\tfrac{3238273}{16777216}+\tfrac{6869081}{8388608}\,i,\qquad a_1=-\tfrac{4425427}{16777216}-\tfrac{256269}{2097152}\,i,\qquad
  a_2=\tfrac{2126359}{8388608}-\tfrac{6838355}{8388608}\,i .
\]
Then
\[
  Q'(z)=18\,(z-1)^5\,\hat s(z)\prod_{j=1}^{10}(z-\rho_j),
\]
with
\begin{align*}
\rho_1 &= \tfrac{524767}{524288}+\tfrac{3303}{1048576}\,i, &
\rho_2 &= \tfrac{262319}{262144}+\tfrac{3785}{1048576}\,i, &
\rho_3 &= \tfrac{510667}{1048576}+\tfrac{596603}{524288}\,i,\\
\rho_4 &= \tfrac{140827}{1048576}+\tfrac{1177373}{1048576}\,i, &
\rho_5 &= -\tfrac{20639}{32768}+\tfrac{439225}{524288}\,i, &
\rho_6 &= -\tfrac{105435}{131072}-\tfrac{652651}{1048576}\,i,\\
\rho_7 &= -\tfrac{291711}{524288}-\tfrac{113639}{131072}\,i, &
\rho_8 &= -\tfrac{96673}{524288}-\tfrac{1084035}{1048576}\,i, &
\rho_9 &= \tfrac{524719}{524288}-\tfrac{3195}{262144}\,i,\\
\rho_{10} &= \tfrac{262337}{262144}-\tfrac{4087}{1048576}\,i .
\end{align*}
Put $Q(z)=\int_b^zQ'(t)\,dt$, so that $Q(b)=0$ and $Q$ is monic of degree $18$. The quadratic $\hat s(z)=z^2+s_1z+s_0$ is fixed by
$Q(a_1)=Q(a_2)=0$: with $\mathcal V(t)=(t-1)^5\prod_j(t-\rho_j)$, $(s_0,s_1)$ is the unique solution of the linear system
\[
  \int_b^{a_i}(t^2+s_1t+s_0)\,\mathcal V(t)\,dt=0\qquad(i=1,2),
\]
and $s_0$, $s_1$ are Gaussian rationals whose real and imaginary parts have denominators of $451$ to $458$ digits; numerically the
zeros of $\hat s$ are $1.00000688\ldots-0.00152261\ldots\,i$ and $-0.41009053\ldots+1.04265219\ldots\,i$. Then
$H(z)=(1-\bar a_1z)(1-\bar a_2z)\,Q(z)/((z-a_1)(z-a_2))$ is a polynomial of degree $18$, and $R(z)=18\,\hat s(z)\prod_j(z-\rho_j)$.

For these data the facts (D), (C), (R) and (S) of the proof of Theorem~\ref{thm:twozeroK2N18} hold; each is verified in exact
arithmetic over $\mathbb Q(i)$; (R) is checked as described after Lemma~\ref{lem:certify}, with the resultant of degree at most $578$ in $c$ computed at
the $581$ integer points $c=0,1,\dots,580$ (as in~\cite[Sect.~5]{GoelShort}). For (C) the division of $Q'$ by $(z-1)^5$ is exact,
$R(1)\neq0$, and Lemma~\ref{lem:SC} is applied to $R$, of degree $12$; for (S) it is applied to $H'$, of degree $17$.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.45,baseline=(current bounding box.center),
    zeroQ/.style={circle,fill,inner sep=1.3pt},
    crit/.style={cross out,draw,thick,inner sep=1.6pt}]
  \draw[gray!40,thin] (-1.25,0)--(1.25,0) (0,-1.2)--(0,1.25);
  \draw[thick] (0,0) circle (1);
  \node[zeroQ,label={[label distance=-1pt]below:$b$}] at (-0.1930,0.8189) {};
  \node[zeroQ,label={[label distance=-1pt]below:$a_1$}] at (-0.2638,-0.1222) {};
  \node[zeroQ,label={[label distance=-1pt]above:$a_2$}] at (0.2535,-0.8152) {};
  \foreach \pt in {(0.4870,1.1379),(0.1343,1.1228),(-0.6299,0.8378),(-0.8044,-0.6224),
                   (-0.5564,-0.8670),(-0.1844,-1.0338),(-0.4101,1.0427)}
    \node[crit] at \pt {};
  \node[crit,very thick,inner sep=3.2pt] at (1,0) {};
  \node[below right] at (1.04,-0.02) {$1$};
\end{tikzpicture}
\hspace{0.05\textwidth}
\includegraphics[width=0.32\textwidth,valign=c]{figures/fig_image.pdf}
\caption{Left: the data of Theorem~\ref{thm:twozeroK2N18}. Dots: the zeros $b$, $a_1$, $a_2$ of $Q$ in $\mathbb D$. Crosses: the $17$
critical points of $Q$, counted with multiplicity; none lies in $\mathbb D$. The large cross at $z=1$ stands for the ten of them
within $0.0123$ of $1$: the zero of order $5$, one zero of $\hat s$ and four of the $\rho_j$. The small crosses are the other seven. Right:
the image $H(\mathbb D)=N_b\,W(\mathbb D)$, computed numerically; axes omitted. The dot is $H(b)=0$, next to the tip of a
narrow inlet whose two sides almost touch.}\label{fig:pair}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[width=0.86\textwidth]{figures/fig_margin.pdf}
\caption{For pair~\ref{pair:K2N18}: how close the zeros of $\mathcal D_c$ come to the unit circle, computed numerically for $0\le c\le1$, on a
logarithmic scale; since $\mathcal D_{-c}(w)=\mathcal D_c(-w)$, the interval $-1\le c\le0$ gives its mirror image. By (R) of Lemma~\ref{lem:certify} the margin
is positive for every $c$. Its lowest point is near $c=0.9231$. At each dip a zero of $\mathcal D_c$ is close to the circle.}\label{fig:margin}
\end{figure}

The zeros of $R$, $H'$ and the $\mathcal D_c$ lie outside $\overline{\mathbb D}$. We call the distance from the unit circle to the nearest zero
of such a polynomial its \emph{margin}\label{term:margin}. The zero of $R$ nearest to the circle, a zero of $\hat s$, has modulus
$1.00000804\ldots$; four of the $\rho_j$
lie within $0.0123$ of $z=1$ as well. The zero of $H'$ nearest to the circle has modulus $1.00001141\ldots$; since
$\mathcal D_{\pm1}(w)=H'(\pm w)$, this gives the margin at $c=\pm1$. Here the smallest margin over $c\in[-1,1]$ is not at $c=\pm1$: numerically
the zeros of the $\mathcal D_c$ come closest near $c=\pm0.9231$, within about $7.028\cdot10^{-6}$ of the circle. Condition (S) concerns $c=\pm1$ only; the other $c$ are covered by (R)
(Figure~\ref{fig:margin}). A small margin has a geometric meaning. For $-1<c<1$ and $c=\cos\theta$, a zero $w$ of $\mathcal D_c$ gives
$H(we^{i\theta})=H(we^{-i\theta})$ with $|w|>1$; so when $|w|$ is close to $1$, two points of the boundary curve of $H(\mathbb D)$
nearly coincide. Numerically, at the dips of Figure~\ref{fig:margin} this happens along the narrow inlet of Figure~\ref{fig:pair}, whose two sides
almost touch.