From 4d8ad50a1393375f561faf7c0ba6eddcd848f990 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Dec 2020 12:25:55 +0100
Subject: Shorted and reflowed some lines.

---
 bezout.tex | 18 ++++++++++++------
 1 file changed, 12 insertions(+), 6 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 10e0167..6fc863c 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -217,7 +217,10 @@ where
     \end{aligned}
   \end{equation}
 \end{widetext}
-This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter vanishes as $a\to1$, where we should recover known results for the real $p$-spin.
+This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot
+z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter
+vanishes as $a\to1$, where we should recover known results for the real
+$p$-spin.
 
 \begin{figure}[htpb]
   \centering
@@ -255,7 +258,8 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of
 \end{equation}
 which makes its ensemble that of Gaussian complex symmetric matrices. Given its variances
 $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
-$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse
+$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
+constant inside the ellipse
 \begin{equation} \label{eq:ellipse}
   \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+
   \left(\frac{\mathop{\mathrm{Im}}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2
@@ -342,7 +346,8 @@ gives
   =(p-1)^N.
 \end{equation}
 This is 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
+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
 \begin{equation} \label{eq:count.def.marginal}
   \overline{\mathcal N}(\kappa,\epsilon)
   =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a)
@@ -376,9 +381,10 @@ contained within is
   =(p-1)^{N/2},
 \end{equation}
 the square root of \eqref{eq:bezout} and precisely the number of critical
-points of the real pure spherical $p$-spin model. (note the role of conjugation symmetry, already underlined in Ref\cite{Bogomolny_1992_Distribution}). In fact, the full
-$\epsilon$-dependence of the real pure spherical $p$-spin is recovered by this
-limit as $\epsilon$ is varied.
+points of the real pure spherical $p$-spin model. (note the role of conjugation
+symmetry, already underlined in Ref\cite{Bogomolny_1992_Distribution}). In
+fact, the full $\epsilon$-dependence of the real pure spherical $p$-spin is
+recovered by this limit as $\epsilon$ is varied.
 
 \begin{figure}[htpb]
   \centering
-- 
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