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.bib | 14 +++++++++++++-
 bezout.tex | 29 ++++++++++++++++++-----------
 2 files changed, 31 insertions(+), 12 deletions(-)

diff --git a/bezout.bib b/bezout.bib
index ae12521..022f5ea 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -5,7 +5,6 @@
  publisher = {Springer Science and Business Media LLC},
  year = {2016},
  month = {12},
- number = {71},
  volume = {2016},
  pages = {71},
  url = {https://doi.org/10.1007%2Fjhep12%282016%29071},
@@ -228,6 +227,19 @@
  subtitle = {Theory and Practice}
 }
 
+@article{Manschot_2012_From,
+ author = {Manschot, Jan and Pioline, Boris and Sen, Ashoke},
+ title = {From black holes to quivers},
+ journal = {Journal of High Energy Physics},
+ publisher = {Springer Science and Business Media LLC},
+ year = {2012},
+ month = {11},
+ volume = {2012},
+ pages = {23},
+ url = {https://doi.org/10.1007%2Fjhep11%282012%29023},
+ doi = {10.1007/jhep11(2012)023}
+}
+
 @article{Nguyen_2014_The,
  author = {Nguyen, Hoi H. and O'Rourke, Sean},
  title = {The Elliptic Law},
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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