From b5521d591037a99233433955a561427c61f74e5f Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Mon, 29 May 2023 17:14:02 +0100
Subject: [PATCH] Simplify expression in theorem 6.1
---
main.tex | 50 ++++++++++++++++++++++++++++++++++----------------
1 file changed, 34 insertions(+), 16 deletions(-)
diff --git a/main.tex b/main.tex
index e282d78..6bbe39e 100644
--- a/main.tex
+++ b/main.tex
@@ -916,28 +916,46 @@ radius of the pseudo-wall being positive
\frac{1}{2n^2}\ZZ
\end{equation}
+\begin{sagesilent}
+var("Delta", domain="real") # placeholder for the specific values of 1/epsilon
+
+r_upper_bound1 = (
+ (1/Delta < bgmlv2_d_upperbound_exp_term)
+ * r * Delta
+)
+
+assert r_upper_bound1.lhs() == r
+
+r_upper_bound2 = (
+ (1/Delta < bgmlv3_d_upperbound_exp_term_alt2)
+ * (r-R) * Delta + R
+)
+
+assert r_upper_bound2.lhs() == r
+\end{sagesilent}
+
\begin{theorem}[Bound on $r$ \#1]
\label{thm:rmax_with_uniform_eps}
Let $v = (R,C,D)$ be a fixed Chern character. Then the ranks of the
- pseudo-semistabilizers for $v$ are bounded above by the following expression.
+ pseudo-semistabilizers for $v$ with
+ $\chern_1^\beta = q$
+ are bounded above by the following expression.
+ \bgroup
+ \def\Delta{\lcm(m,2n^2)}
+ \def\psi{\chern_1^{\beta}(F)}
\begin{align*}
- &\frac{\lcm(m,2n^2)}{2}
- \max_{q \in [0,\chern_1^\beta(v)]}
- \\
- &\left\{
- \min
- \left(
- q^2,
- 2R\beta q
- +C^2
- -2DR
- -2Cq
- +q^2
- +\frac{R}{\lcm(m,2n^2)}
- \right)
- \right\}
+ \min
+ \left(
+ \sage{r_upper_bound1.rhs()}, \:\:
+ \sage{r_upper_bound2.rhs()}
+ \right)
\end{align*}
+ \egroup
+
+ Taking the maximum of this expression over
+ $q \in [0, \chern_1^{\beta}(F)]\cap \frac{1}{n}\ZZ$
+ would give an upper bound for the ranks of pseudo-semistabilizers for $v$.
\end{theorem}
\begin{proof}
--
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