From 43c063ded0e682f7f8d81e2c8ba48fdab1e27548 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Mon, 15 May 2023 00:19:00 +0100
Subject: [PATCH] Add theorem for final bound for rmax
---
main.tex | 24 ++++++++++++++++++++++++
1 file changed, 24 insertions(+)
diff --git a/main.tex b/main.tex
index 489bee5..2d142d8 100644
--- a/main.tex
+++ b/main.tex
@@ -993,6 +993,7 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
Finding better alternatives to $\epsilon_F$:
$\epsilon_q^1$ and $\epsilon_q^2$
]
+\label{lemdfn:epsilon_q}
Suppose $d \in \frac{1}{m}\ZZ$ satisfies the condition in
eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
That is:
@@ -1085,6 +1086,29 @@ which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_q^2$.
\end{proof}
+\begin{theorem}[Bound on $r$ \#3]
+\label{thm:rmax_with_eps1}
+ Let $v = (R,C,D)$ be a fixed Chern character. Then the ranks of the
+ pseudo-semistabilizers for $v$ with
+ $\chern_1^\beta = q = \frac{a_q}{n}$
+ are bounded above by the following expression (with $i=1$ or 2).
+
+ \begin{equation*}
+ \frac{1}{2 \epsilon_q^i}
+ \min
+ \left(
+ q^2,
+ 2R\beta q
+ +C^2
+ -2DR
+ -2Cq
+ +q^2
+ +\frac{R}{\lcm(m,2n^2)}
+ +R \epsilon_q^i
+ \right)
+ \end{equation*}
+ Where $\epsilon_q^i$ is defined as in definition/lemma \ref{lemdfn:epsilon_q}.
+\end{theorem}
\minorheading{Irrational $\beta$}
--
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