Update on Overleaf.

1 files changed, 9 insertions, 7 deletions

diff --git a/bezout.tex b/bezout.tex
index 5f55300..1c416e2 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -30,7 +30,7 @@
   solutions averaged over randomness in the $N\to\infty$ limit.  We find that
   it saturates the Bézout bound $\log\overline{\mathcal{N}}\sim N \log(p-1)$.
   The Hessian of each saddle is given by a random matrix of the form $C^\dagger
-  C$, where $C$ is a complex Gaussian matrix with a shift to its diagonal.  Its
+  C$, where $C$ is a complex {\color{red} symmetric} Gaussian matrix with a shift to its diagonal.  Its
   spectrum has a transition where a gap develops that generalizes the notion of
   `threshold level' well-known in the real problem.  The results from the real
   problem are recovered in the limit of real parameters. In this case, only the
@@ -95,7 +95,7 @@ introducing $\epsilon\in\mathbb C$, our energy is
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
-At any critical point, $\epsilon=H/N$, the average energy. We choose to
+ We choose to
 constrain our model by $z^2=N$ rather than $|z|^2=N$ in order to preserve the
 analyticity of $H$. The nonholomorphic constraint also has a disturbing lack of
 critical points nearly everywhere: if $H$ were so constrained, then
@@ -104,22 +104,24 @@ $0=\partial^* H=-p\epsilon z$ would only be satisfied for $\epsilon=0$.
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation}
   \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
+  = p\epsilon z_i \label{cosa}
 \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.  In this sense
+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 (\ref{cosa}) 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
-therefore the same as that of $\operatorname{Re}H$. From each critical point
+therefore the same as that of $\operatorname{Re}H$. From each saddle 
 emerges a gradient line of $\operatorname{Re}H$, which is also one of constant
 $\operatorname{Im}H$ and therefore constant phase.
 
 Writing $z=x+iy$, $\operatorname{Re}H$ can be considered a real-valued function
-of $2N$ real variables. Its number of critical points is given by the usual
+of $2N$ real variables. Its number of saddle-points is given by the usual
 Kac--Rice formula:
 \begin{equation} \label{eq:real.kac-rice}
   \begin{aligned}
@@ -152,7 +154,7 @@ transformations through, we have
 \end{equation}
 This gives three equivalent expressions for the determinant of the Hessian: as
 that of a $2N\times 2N$ real matrix, that of an $N\times N$ Hermitian matrix,
-or the norm squared of that of an $N\times N$ complex symmetric matrix.
+i.e. the norm squared of that of an $N\times N$ complex symmetric matrix.
 
 These equivalences belie a deeper connection between the spectra of the
 corresponding matrices. Each positive eigenvalue of the real matrix has a