From f7df7235a9b8d44ab47a61aec32b215c88549fa1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 12:40:58 +0100
Subject: Spot changes.

---
 bezout.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index efe9b62..c0ecff7 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -103,11 +103,11 @@ a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$. The reasoning
 behind the choice is twofold.
 
 First, we seek draw conclusions from our model that would be applicable to
-generic holomorphic functions without any symmetry.  Samples of $H_0$ nearly
+generic holomorphic functions without any symmetry. Samples of $H_0$ nearly
 provide this, save for a single anomaly: the value of the energy and its
 gradient at any point $z$ correlate along the $z$ direction, with
 $\overline{H_0\partial_iH_0}\propto\kappa(z^2)^{p-1}z_i$ and
-$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$.  Besides being a
+$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$. Besides being a
 spurious correlation, in each sample there is also a `radial' gradient of
 magnitude proportional to the energy, since $z\cdot\partial H_0=pH_0$. This
 anomalous direction must be neglected if we are to draw conclusions about
@@ -132,7 +132,7 @@ equations of degree $p-1$, to which one must add the constraint.  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 \eqref{eq:polynomial} that at any critical
-point, $\epsilon=H/N$, the average energy.
+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
@@ -187,7 +187,7 @@ $(\partial\partial H)^\dagger\partial\partial H$.
 
 The expression \eqref{eq:complex.kac-rice} is to be averaged over $J$ to give
 the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N} = \int dJ \,
-\log \mathcal N_J$, a calculation that involves the replica trick. In most the
+\log \mathcal N_J$, a calculation that involves the replica trick. In most of the
 parameter-space that we shall study here, the \emph{annealed approximation} $N
 \Sigma \sim \log \overline{ \mathcal N} = \log\int dJ \, \mathcal N_J$ is
 exact. 
-- 
cgit v1.2.3-54-g00ecf

