From f621928ba5d582998232b1c26b4eb48424fa77eb Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 21 Jun 2023 20:05:56 +0100
Subject: [PATCH] Rewrite second reccurring ex more directly
---
main.tex | 14 +++++++-------
1 file changed, 7 insertions(+), 7 deletions(-)
diff --git a/main.tex b/main.tex
index ae1ebb9..a0ce7aa 100644
--- a/main.tex
+++ b/main.tex
@@ -1483,14 +1483,14 @@ that $m=2$, $\beta=\sage{recurring.b}$,
giving $n=\sage{recurring.b.denominator()}$.
\begin{sagesilent}
-n = recurring.b.denominator()
-m = 2
+recurring.n = recurring.b.denominator()
recurring.bgmlv = recurring.chern.Q_tilt()
-recurring.lcm = lcm(m, 2*n^2)
corrolary_bound = (
- recurring.bgmlv * recurring.lcm / 8
- + recurring.chern.ch[0] / 2
- + recurring.chern.ch[0]^2 / (2*recurring.bgmlv*recurring.lcm)
+ r_upper_bound_all_q.expand()
+ .subs(Delta==recurring.bgmlv)
+ .subs(nu==1) ## \ell^2=1 on P^2
+ .subs(R==recurring.chern.ch[0])
+ .subs(n==recurring.n)
)
\end{sagesilent}
Using the above corrolary \ref{cor:direct_rmax_with_uniform_eps}, we get that
@@ -1506,7 +1506,7 @@ Furthermore, the proof of theorem \ref{thm:rmax_with_uniform_eps} uses the fact
that, given an element of $\frac{1}{2n^2}\ZZ$, the closest non-equal element of
$\frac{1}{2}\ZZ$ is at least $\frac{1}{2n^2}$ away. However this a
conservative estimate, and a larger gap can sometimes be guaranteed if we know
-this value of $\frac{1}{2n^2}\ZZ$ explicitly.
+this value of $\frac{1}{2n^2}\ZZ$ explicitly.
The expressions that will follow will be a bit more complicated and have more
parts which depend on the values of $q$ and $\beta$, even their numerators
--
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