From 4ffbe5812811c329cbd6674b472bf4cc9ba581a1 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Thu, 10 Dec 2020 13:31:03 +0000
Subject: Update on Overleaf.

---
 bezout.tex | 11 +++++------
 1 file changed, 5 insertions(+), 6 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 4c4ff46..7da708b 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -92,8 +92,7 @@ not possible. The same idea may be implemented by performing diffusion in the
 $J$'s, and following the roots, in complete analogy with Dyson's stochastic
 dynamics.
 
-This study also provides a complement to the work on the distribution of zeroes
-of random polynomials \cite{Bogomolny_1992_Distribution}.
+
 
 
   For our model the  constraint we  choose  $z^2=N$,
@@ -113,11 +112,11 @@ Critical points are given by the set of equations:
 \begin{equation}
 \frac{c_p}{(p-1)!}\sum_{ i, i_2\cdots i_p}^NJ_{i, i_2\cdots i_p}z_{i_2}\cdots z_{i_p} = \epsilon z_i
 \end{equation}
-which for given $\epsilon$ are a set pf $N$ equations (plus the constraint) of degree $p-1$.
+which for given $\epsilon$ are a set of $N$ equations of degree $p-1$, to which one must add the constraint condition.
+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 $p \rightarrow \infty$.
 
-$
-Since $H$ is holomorphic, a point is a critical point of its real part if and
-only if it is also a critical point of its imaginary part. The number of
+Since $H$ is holomorphic, a critical point of $\Re H_0$ is also a critical  point of $\Im H_0$. The number of
 critical points of $H$ is therefore the number of critical points of
 $\mathop{\mathrm{Re}}H$. Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be
 interpreted as a real function of $2N$ real variables. The number of critical
-- 
cgit v1.2.3-54-g00ecf

