Add subscripts to a and b to denote what they depend on

main.tex

@@ -781,22 +781,24 @@ Some of the details around the associated numerics are explored next.
 
 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 $n \in \NN,a \in \ZZ$.
+\renewcommand{\aa}{{a_F}}
+\newcommand{\bb}{{b_q}}
+Suppose $\beta = \frac{\aa}{n}$ for some coprime $n \in \NN,\aa \in \ZZ$.
 Then fix a value of $q$:
 \begin{equation}
 	q:=\chern_1^{\beta}(E)
-	  =\frac{b}{n}
+	  =\frac{\bb}{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 only consider $r$-values for which $c:=\chern_1(E)$ is not integral:
+Firstly, we only consider $r$-values for which $c:=\chern_1(E)$ is 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)
+var("a_F b_q n") # Define symbols introduce for values of beta and q
+beta_value_expr = (beta == a_F/n)
+q_value_expr = (q == b_q/n)
 \end{sagesilent}
 
 \begin{equation}
@@ -806,7 +808,8 @@ beta_value_expr = (beta == a/n)
 \end{equation}
 
 \noindent
-That is, $r \equiv -a^{-1}b$ mod $n$ ($a$ is coprime to $n$, and so invertible mod $n$).
+That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
+$n$, and so invertible mod $n$).
 
 Substituting the current values of $q$ and $\beta$ into the condition for the
 radius of the pseudo-wall being positive
@@ -816,7 +819,9 @@ radius of the pseudo-wall being positive
 \label{eqn:positive_rad_condition_in_terms_of_q_beta}
 	\frac{1}{m}\ZZ
 	\ni
+	\qquad
 	\sage{positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()}
+	\qquad
 	\in
 	\frac{1}{2n^2}\ZZ
 \end{equation}
@@ -838,7 +843,7 @@ this happens when:
 		\sage{bgmlv2_d_upperbound_exp_term},
 		\sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)},
 	\right)
-	< \epsilon := \frac{1}{\lcm(m,2n^2)}
+	< \epsilon_{\beta} := \frac{1}{\lcm(m,2n^2)}
 \end{equation}
 
 %% refinements using specific values of q and beta
@@ -859,13 +864,13 @@ Considering the numerator of the right-hand-side of
 
 \begin{align}
 	\sage{rhs_numerator}
-	&\equiv (a(-a^{-1}b)+2b)a &\mod n
+	&\equiv (\aa(-\aa^{-1}\bb)+2\bb)\aa &\mod n
 \\
-	&\equiv ab &\mod n
+	&\equiv \aa\bb &\mod n
 \end{align}
 
 \noindent
-And so, we also have $a(ar+2b) \equiv ab$ (mod $2n^2$).
+And so, we also have $\aa(\aa r+2\bb) \equiv \aa\bb$ (mod $2n^2$).
 
 
 \minorheading{Irrational $\beta$}