Add restriction on r for c to be integral

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}