From 57d6f4d6fd0a7fdb738b91354de063f6a74847d1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 12:30:49 +0100
Subject: More rewriting of constraint reasoning.

---
 bezout.tex | 34 ++++++++++++++++++++++------------
 1 file changed, 22 insertions(+), 12 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 7cff5f1..efe9b62 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -99,18 +99,28 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
 One might balk at the constraint $z^2=N$---which could appropriately be called
-a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$.  The reasoning
-is twofold. First, at every point $z$ the energy \eqref{eq:bare.hamiltonian}
-has a `radial' gradient of magnitude proportional to the energy, since
-$z\cdot\partial H_0=pH_0$. This anomalous direction must be neglected if critical
-points are to exist at any nonzero energy, and the constraint surface $z^2=N$ is
-the unique surface whose normal is parallel to $z$ and which contains the
-configuration space of the real $p$-spin model as a subspace. Second, taking
-the constraint to be the level set of a holomorphic function means the
-resulting configuration space is a \emph{bone fide} complex manifold, and
-therefore permits easy generalization of the integration techniques referenced
-above. The same cannot be said for the space defined by $|z|^2=N$, which is
-topologically the $(2N-1)$-sphere and cannot admit a complex structure.
+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
+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
+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
+generic functions, and the constraint surface $z^2=N$ is the unique surface
+whose normal is parallel to $z$ and which contains the configuration space of
+the real $p$-spin model as a subspace.
+
+Second, taking the constraint to be the level set of a holomorphic function
+means the resulting configuration space is a \emph{bone fide} complex manifold,
+and therefore permits easy generalization of the integration techniques
+referenced above. The same cannot be said for the space defined by $|z|^2=N$,
+which is topologically the $(2N-1)$-sphere and cannot admit a complex
+structure.
 
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
-- 
cgit v1.2.3-54-g00ecf

