From 48907aaba8f23eadfc51f07795f5053944d5a502 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Fri, 12 May 2023 16:16:26 +0100
Subject: [PATCH] Add restriction on r for c to be integral
---
main.tex | 50 ++++++++++++++++++++++++++++++++++++++++----------
1 file changed, 40 insertions(+), 10 deletions(-)
diff --git a/main.tex b/main.tex
index 6b844a9..f2fd268 100644
--- a/main.tex
+++ b/main.tex
@@ -33,6 +33,7 @@ Practical Methods for Finding Pseudowalls}
\maketitle
\section{Introduction}
+\label{sec:intro}
[ref] shows that for any rational $\beta_0$,
the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
@@ -81,6 +82,7 @@ bounds on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
Characters}
\section{Twisted Chern Characters of Pseudo Destabilizers}
+\label{sec:twisted-chern}
For a given $\beta$, we can define a twisted Chern character
$\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)$:
@@ -139,6 +141,7 @@ for the rank of $E$:
\section{Limitations}
\section{Refinement}
+\label{sec:refinement}
To get tighter bounds on the rank of destabilizers $E$ of some $F$ with some
fixed Chern character, we will need to consider each of the values which
@@ -756,16 +759,9 @@ These lines have the same assymptote at $r \to \infty$
(eqns \ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:positive_rad_d_bound_betamin}).
-The finiteness of these solutions will be entirely determined by whether $\beta$
-is rational or irrational, as covered next.
-
-
-\minorheading{Rational $\beta$}
-
-Suppose $\beta = \frac{*}{n}$ for some $n \in \NN,* \in \ZZ$.
-
-\minorheading{Irrational $\beta$}
-
+As mentioned in the introduction (sec \ref{sec:intro}), the finiteness of these
+solutions is entirely determined by whether $\beta$ is rational or irrational.
+Some of the details around the associated numerics are explored next.
\begin{figure}
\centering
@@ -780,6 +776,40 @@ Suppose $\beta = \frac{*}{n}$ for some $n \in \NN,* \in \ZZ$.
\label{fig:d_bounds_xmpl_gnrc_q}
\end{figure}
+
+\minorheading{Rational $\beta$}
+
+The strategy here is similar to what was shown in (sect \ref{sec:twisted-chern}),
+% ref to Schmidt?
+Suppose $\beta = \frac{a}{n}$ for some coprime $n \in \NN,a \in \ZZ$.
+Then fix a value of $q$:
+\begin{equation}
+ q:=\chern_1^{\beta}(E)
+ =\frac{b}{n}
+ \in
+ \frac{1}{n} \ZZ
+ \cap [0, \chern_1^{\beta}(F)]
+\end{equation}
+as noted at the beginning of this section (\ref{sec:refinement}).
+Firstly, we can ignore $r$-values for which $c:=\chern_1(E)$ is not integral:
+
+\begin{sagesilent}
+var("a b n") # Define symbols introduce for values of beta and q
+q_value_expr = (q == b/n)
+beta_value_expr = (beta == a/n)
+\end{sagesilent}
+
+\begin{equation}
+ c =
+ \sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
+ \in \ZZ
+\end{equation}
+
+That is, $r \equiv -a^{-1}b$ mod $n$ ($a$ is coprime to $n$, and so invertible mod $n$).
+
+\minorheading{Irrational $\beta$}
+
+
\egroup
\section{Conclusion}
--
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