From a88e086debc8d234f1120f8426384d06408b85fe Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 18 Dec 2020 15:44:27 +0100
Subject: Added note and citation.

---
 bezout.tex | 29 ++++++++++++++++++-----------
 1 file changed, 18 insertions(+), 11 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 45820c4..d7f6399 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -103,9 +103,9 @@ critical points nearly everywhere: if $H$ were so constrained, then
 $0=\partial^* H=-p\epsilon z$ would only be satisfied for $\epsilon=0$.  
 
 The critical points are of $H$ given by the solutions to the set of equations
-\begin{equation}
+\begin{equation} \label{eq:polynomial}
   \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}}
-  = p\epsilon z_i \label{cosa}
+  = p\epsilon z_i
 \end{equation}
 for all $i=\{1,\ldots,N\}$, which for fixed $\epsilon$ is a set of $N$
 equations of degree $p-1$, to which one must add the constraint. 
@@ -113,7 +113,7 @@ In this sense
 this study also provides a complement to the work on the distribution of zeroes
 of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$
 and $p\to\infty$.
-We see from (\ref{cosa}) that at any critical point, $\epsilon=H/N$, the average energy.
+We see from \eqref{eq:polynomial} that at any critical point, $\epsilon=H/N$, the average energy.
 
 Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a
 critical point of $\operatorname{Im}H$. The number of critical points of $H$ is
@@ -339,13 +339,20 @@ values of $\kappa$ and $\epsilon$. Taking this saddle gives
   \log\overline{\mathcal N}(\kappa,\epsilon)
   =N\log(p-1).
 \end{equation}
-This is, to this order,  precisely the Bézout bound, the maximum number of solutions that $N$
-equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is
-gained by looking at the count as a function of $a$, defined by $\overline{\mathcal
-N}(\kappa,\epsilon,a)=e^{Nf(a)}$.  In the large-$N$ limit, this is the
-cumulative number of critical points, or the number of critical points $z$ for
-which $|z|^2\leq a$. We likewise define the $a$-dependant complexity
-$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}(\kappa,\epsilon,a)$
+This is, to this order,  precisely the Bézout bound, the maximum number of
+solutions that $N$ equations of degree $p-1$ may have
+\cite{Bezout_1779_Theorie}. That we saturate this bound is perhaps not
+surprising, since the coefficients of our polynomial equations
+\eqref{eq:polynomial} are complex and have no symmetries. Analogous asymptotic
+scaling has been found for the number of pure Higgs states in supersymmetric
+quiver theories \cite{Manschot_2012_From}.
+
+More insight is gained by looking at the count as a function of $a$, defined by
+$\overline{\mathcal N}(\kappa,\epsilon,a)=e^{Nf(a)}$.  In the large-$N$ limit,
+this is the cumulative number of critical points, or the number of critical
+points $z$ for which $|z|^2\leq a$. We likewise define the $a$-dependant
+complexity $\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal
+N}(\kappa,\epsilon,a)$
 
 \begin{figure}[htpb]
   \centering
@@ -443,7 +450,7 @@ threshold level, where the system develops a mid-spectrum gap, will play a
 crucial role as it does in the real case.
 
 \begin{acknowledgments}
-We wish to thank Alexander Altland, Satya Majumdar and Gregory Schehr for a useful suggestions.
+  We wish to thank Alexander Altland, Satya Majumdar and Gregory Schehr for a useful suggestions.
   JK-D and JK are supported by the Simons Foundation Grant No.~454943.
 \end{acknowledgments}
 
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