From 487e9b69a43428aa1f8a19e0a16c228509e21786 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Fri, 11 Dec 2020 09:42:52 +0000
Subject: Update on Overleaf.

---
 bezout.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 1c416e2..fa973cc 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -336,7 +336,7 @@ values of $\kappa$ and $\epsilon$. Taking this saddle gives
   \log\overline{\mathcal N}(\kappa,\epsilon)
   =N\log(p-1).
 \end{equation}
-This is precisely the Bézout bound, the maximum number of solutions that $N$
+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
@@ -364,7 +364,7 @@ Notice that the limit of this expression as $a\to\infty$ corresponds with
 several values of $\kappa$ in Fig.~\ref{fig:complexity}. For any $|\kappa|<1$,
 the complexity goes to negative infinity as $a\to1$, i.e., as the spins are
 restricted to the reals.  This is natural, given that the $y$ contribution to
-the volume shrinks to zero as that of an $N$-dimensional sphere with volume
+the volume shrinks to zero as that of an $N$-dimensional sphere $\sum_i y_i^2=N(a-1)$ with volume
 $\sim(a-1)^N$.  However, when the result is analytically continued to
 $\kappa=1$ (which corresponds to real $J$) something novel occurs: the
 complexity has a finite value at $a=1$. Since the $a$-dependence gives a
-- 
cgit v1.2.3-54-g00ecf

