%! TEX root = SGT.tex
% vim: tw=80
% 28/05/2025 11AM

Last time, when working on $\det(xI - A_s)$, we showed:

\begin{fcthm}[Godsil-Gutman, 80s]
  % Theorem 2
  \[
    \Ebb_{S \sim \{\pm 1\}} \det(xI - A_s)
    = \mu_G(x)
  ,\]
  where
  \[
    \mu_G(x)
    = \sum_{k \ge 0} x^{n - 2k} (-1)^k m_k(G)
  ,\]
  where $m_k(G)$ is the number of matchings in $G$ with $k$ edges.
\end{fcthm}

Fact: $\mu_G(x)$ is real rooted.

\begin{fcthm}[Heilman-Lieb, 72]
  \label{thm:heilman3}
  % Theorem 3
  Assuming:
  - $G$ is $d$-regular
  Then:
  \[
    \maxroot \mu_G(x)
    \le 2\sqrt{d - 1}
  .\]
\end{fcthm}

If only we could say that $\maxroot \Ebb_S \det(xI - A_s)$ is an average of
$\maxroot \det(xI - A_h)$, $h \in \{\pm 1\}^E$.
Hopelessly false:
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/daf57239a1904212.png}
\end{center}

\begin{fcdefn}[Interlacing]
  % Definition 4
  Let $f$ be a real rooted polynomial of degree $n$ with roots $\alpha_1,
  \ldots, \alpha_n$ and $g$ a real rooted polynomial of degree $n - 1$ with
  roots $\beta_1, \ldots, \beta_{n - 1}$.
  We say $g$ \emph{interlaces} $f$ if
  \[
    \alpha_1
    \le \beta_1
    \le \alpha_2
    \le \cdots
    \le \beta_{n - 2}
    \le \alpha_{n - 1}
    \le \beta_{n - 1}
    \le \alpha_n
  .\]
\end{fcdefn}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/e562147cfe38493f.png}
\end{center}

\begin{example*}
  If $f$ is real rooted, then so is $f'$, and also $f'$ interlaces $f$.
\end{example*}

\begin{fcdefn}[Common interlacement]
  % Definition 5
  We say that real rooted polynomials $f$, $g$ of degree $n$ have a \emph{common
  interlacement} if there is a polynomial $h$ of degree $n + 1$ such that $f$
  and $g$ both interlace $h$.
  In other words, if the roots of $f$ and $g$ are $\alpha_1 \le \cdots \alpha_n$
  and $\beta_1 \le \cdots \le \beta_n$ respectively, then they have a common
  interlacement if and only if there are some $\gamma_0 \le \cdots \le \gamma_n$
  such that
  \[
    \gamma_0
    \le \alpha_1, \beta_1
    \le \gamma_1
    \le \alpha_2, \beta_2
    \le \gamma_2
    \le \cdots
    \le \gamma_{n - 1}
    \le \alpha_n, \beta_n
    \le \gamma_n
  .\]
\end{fcdefn}

\begin{fcthm}[Fell 80]
  Assuming:
  - $f$, $g$ are real rooted
  - both degree $n$, monic
  Then:
  $f$ and $g$ have a common interlacing if and only if every convex combination
  of $f$ and $g$ are real rooted.
\end{fcthm}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/96c4953a5aa54504.png}
\end{center}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/1b75cf48fdaa491a.png}
\end{center}

\begin{proof}
  Assume $f$ and $g$ without repeated or common roots (general case requires
  more work and details checking).
  \begin{enumerate}[$\Rightarrow$]
    \item[$\Leftarrow$] Let $h_t(x) = tf(x) + (t - 1) g(x)$ for $t \in (0, 1)$.
      We know these are real rooted, and waht to show that $f$, $g$ have a
      common interlacement.
      Let $\lambda_i(t)$ be the $i$-th root of $h_t(x)$.
      $\lambda_i(t)$ includes $(\lambda_i(0), \lambda_i(1)) \subseteq \Rbb$.

      Claim: $(\lambda_I(0), \lambda_I(1))$ are disjoint.

      Suppose not.
      Let $\lambda_k$ be a root of $f$ and $\lambda_k \in (\lambda_I(0),
      \lambda_I(1))$.
      So there exists $t \in (0, 1)$ such that $\lambda_I(t) = \lambda_k$.
      \[
        h_t(\lambda_k)
        = 0
        = tf(\lambda_k) + (1 - t) g(\lambda_k)
      ,\]
      so $g(\lambda_k) = 0$, contradiction as we assumed no common roots.
    \item[$\Rightarrow$]
      \leavevmode
      \begin{center}
        \includegraphics[width=0.6\linewidth]{images/9de811e8b18c45f0.png}
      \end{center}
      The dots represent the roots of the polynomial that interlaces them.
      By monicness, we get the blue and green $+$s and $-$s.
      Then get the orange ones, and use intermediate value theorem.
      \qedhere
  \end{enumerate}
\end{proof}

\begin{fccoro}[]
  % Corollary 7
  Assuming:
  - $f$, $g$ are real rooted of degree $n$
  - have a common interlacing
  Then:
  for all $t \in [0, 1]$
  \[
    \min \{\maxroot(f), \maxroot(g)\}
    \le \maxroot(tf(x) + (1 - t)g(x))
  .\]
\end{fccoro}

Approach to \cref{thm:exists signing}:

$f(x) = \Ebb_{S \in \{\pm 1\}^E} \det(xI - A_s)$, $|E| = m$.
For $h_1, \ldots, h_k \in \{\pm 1\} = \{+, -\}$, then
\[
  f_{h_1, \ldots, h_k}(x)
  = \Ebb_{s_{k + 1}, \ldots, s_m \in \{\pm 1\}} (\det(xI - A_s) \mid s_1 = h_1,
  \ldots, s_k = h_k)
.\]

``Method of interlacing families of polynomials''
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/f563eaeb15b240db.png}
\end{center}

\begin{fcthm}[Marcus, Spielman, Srivastava (real rooted)]
  % THeorem 8
  \label{thm:real rooted}
  For every $\rho_1, \ldots, \rho_m \in [-1, 1]$ the following polynomial is
  real rooted 
  \[
    \chi_{\rho_1, \ldots, \rho_m}(x)
    = \sum_{S \in \{\pm 1\}^m} \det(xI - A_s)
    \prod_{J : s_J = 1} \left( \frac{1 + \rho_J}{2} \right)
    \prod_{J : S_J = -1} \left( \frac{1 - \rho_J}{2} \right)
  .\]
\end{fcthm}

\cref{thm:real rooted} implies \cref{thm:exists signing}:
\[
  f_{h_1, \ldots, h_k}(x)
  = \chi_{h_1, \ldots, h_k, 0, 0, \ldots, 0}(x)
.\]
\[
  \mu_G(x)
  = \chi_{0, \ldots, 0}(x)
.\]
\[
  tf_{h_1, \ldots, h_k, 1}(x) + (1 - t) f_{h_1, \ldots, h_k, -1}(x)
  = \chi_{h_1, \ldots, h_k, 2t - 1, 0, \ldots, 0}(x)
.\]

\begin{fcthm}[Marcus, Spielman, Srivastava (matrix)]
  \label{thm:matrix}
  % Theorem 9
  Assuming:
  - $r_1, \ldots, r_m \in \Rbb^n$ are independent random vectors, whose support
  has $\le 2$ points
  Then:
  \[
    \Ebb \det \left( xI - \sum_{j = 1}^{m} r_j r_j^\top \right)
  \]
  is real rooted.
\end{fcthm}

\cref{thm:matrix} implies \cref{thm:real rooted}:

$\chi_{\rho_1, \ldots, \rho_m}(x)$ is
\[
  \Ebb \det(xI - A_s)
.\]
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/52728aa3555148d3.png}
\end{center}
$\Ebb \det(xI - A_s)$ is real rooted if and only if $\Ebb \det((x + d)I - A_s)$
is real rooted.
This equals
\[
  \Ebb \det(xI + (dI - A_s))
.\]
Note
\[
  dI - A_S
  = \sum_{x \sim y} (e_x + S(x, y) e_y)(e_x + S(x, y) e_y)^\top
.\]
Done.