From fd5e8f3ad46b2beeac63cb7c3af8a90d8fa2db81 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 30 May 2023 16:12:20 +0100
Subject: [PATCH] Add statement to corrolary of 'bound on r 1' (real
alternative to Schmidt's bound)
---
main.tex | 27 +++++++++++++++++++++++++++
1 file changed, 27 insertions(+)
diff --git a/main.tex b/main.tex
index 60aea11..4fb64de 100644
--- a/main.tex
+++ b/main.tex
@@ -24,6 +24,7 @@
\newcommand{\minorheading}[1]{{\noindent\normalfont\normalsize\bfseries #1}}
\newtheorem{theorem}{Theorem}[section]
+\newtheorem{corrolary}{Corrolary}[section]
\newtheorem{lemmadfn}{Lemma/Definition}[section]
\newtheorem{dfn}{Definition}[section]
@@ -1035,6 +1036,32 @@ This is equivalent to:
\end{proof}
+\begin{corrolary}[Bound on $r$ \#2]
+\label{cor:direct_rmax_with_uniform_eps}
+ Let $v$ be a fixed Chern character and
+ $R:=\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2)\Delta(v)$.
+ Then the ranks of the pseudo-semistabilizers for $v$
+ are bounded above by the following expression.
+
+ \bgroup
+ \begin{equation}
+ \frac{1}{2} \lcm(m,2n^2)
+ \left(
+ \frac{\chern_1^{\beta}(v)}{2}
+ + \frac{R}{\chern_1^{\beta}(v)\lcm(m,2n^2)}
+ \right)^2
+ \end{equation}
+ \egroup
+ \bgroup
+ \begin{equation}
+ \frac{1}{8}
+ \Delta(v) \lcm(m,2n^2)
+ + \frac{1}{2} R
+ + \frac{R^2}{ 2 \Delta(v) \lcm(m,2n^2) }
+ \end{equation}
+ \egroup
+\end{corrolary}
+
%% TODO simplified expression for rmax by predicting which q gives rmax
%% refinements using specific values of q and beta
--
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