From fb98534843c67fa994657a9628f3f315856f3ece Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 7 Sep 2024 15:19:51 +0200
Subject: Retooled the abstract.

---
 topology.tex | 31 +++++++++++++++++--------------
 1 file changed, 17 insertions(+), 14 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index a401a65..6252c96 100644
--- a/topology.tex
+++ b/topology.tex
@@ -52,20 +52,23 @@ $\star$ \href{mailto:jaron.kent-dobias@roma1.infn.it}{\small jaron.kent-dobias@r
 
 \section*{\color{scipostdeepblue}{Abstract}}
 \textbf{\boldmath{%
-We consider the set of solutions to $M$ random polynomial equations with
-independent Gaussian coefficients and a target value $V_0$ on the $(N-1)$-sphere. When solutions
-exist, they form a manifold. We compute the average Euler characteristic of
-this manifold in the limit of large $N$, and find different behavior
-depending on the ratio $\alpha=M/N$. When $\alpha<\alpha_\text{onset}$, the
-average characteristic is 2 and there is a single connected component, while
-for $\alpha_\text{onset}<\alpha<\alpha_\text{shatter}$ many large connected components coexist. When
-$\alpha>\alpha_\text{shatter}$ the large connected components vanish, replaced by small fragments, and the
-entire manifold vanishes for $\alpha>\alpha_\text{\textsc{sat}}$. In the
-limit $\alpha\to0$ there is a correspondence between this problem and the
-topology of constant-energy level sets in the spherical spin glasses. We
-conjecture that the energy $E_\text{shatter}$ associated with the vanishing of
-the large connected component corresponds to the asymptotic limit of gradient
-descent from a random initial condition.
+We consider the set of solutions to $M$ random polynomial equations of $N$
+variables. Each equation has independent Gaussian coefficients and a target
+value $V_0$, and their variables are restricted to the $(N-1)$-sphere. When
+solutions exist, they form a manifold. We compute the average Euler
+characteristic of this manifold in the limit of large $N$, and find different
+behavior depending on the target value $V_0$, the ratio $\alpha=M/N$, and the
+variances of the polynomial coefficients. We divide the behavior in four
+phases: a connected phase, an onset phase, a shattered phase, and an
+\textsc{unsat} phase. In the connected phase, the average characteristic is 2
+and there is a single extensive connected component, while in the onset phase
+the Euler characteristic is exponentially large in $N$. In the shattered phase
+the characteristic remains exponentially large but subextensive components
+appear, while in the \textsc{unsat} phase the manifold vanishes. When $M=1$
+there is a correspondence between this problem and the topology of
+energy level sets in the spherical spin glasses. We conjecture that
+the transition from the onset to shattered phase corresponds to the asymptotic
+limit of gradient descent from a random initial condition.
 }
 }
 
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