main.tex

(or corollary \ref{cor:num_test_prob2})

\subsubsection{Size of pseudo-wall: $\chern_2^P(u)>0$ }
\label{subsect-d-bound-radiuscond}

This condition refers to condition
\ref{item:radiuscond:lem:num_test_prob1}
from lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).

In the case where we are tackling problem \ref{problem:problem-statement-2}
(with $\beta = \beta_{-}$), this condition, when expressed as a bound on $d$,
amounts to:

\begin{align}
\label{eqn:radius-cond-betamin}
	\chern_2^{\beta_{-}}(u) &> 0 \\
	d &> \beta_{-}q + \frac{1}{2} \beta_{-}^2r
\end{align}

\begin{sagesilent}
import other_P_choice as problem1
\end{sagesilent}

In the case where we are tackling problem \ref{problem:problem-statement-1},
with some Chern character $v$ with positive rank, and some choice of point
$P=(A,B) \in \Theta_v^-$.
Then $\sage{problem1.A2_subs}$ follows from $\chern_2^P(v)=0$. Using this substitution into the
condition $\chern_2^P(u)>0$ yields:

\begin{equation}
	\sage{problem1.radius_condition}
\end{equation}

\noindent
Expressing this as a bound on $d$, then yields:

\begin{equation}
	\sage{problem1.radius_condition_d_bound}
\end{equation}


\subsubsection{
	Semistability of the Semistabilizer:
	\texorpdfstring{
		$\Delta(u) \geq 0$
	}{
		Δ(u) ≥ 0
	}
}
This condition refers to condition
\ref{item:bgmlvu:lem:num_test_prob1}
from lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).


\noindent
Expressing $\Delta(u)\geq 0$ in terms of $q$ as defined in eqn \ref{eqn-cintermsofm}
we get the following:


\begin{sagesilent}
from plots_and_expressions import bgmlv2_with_q
\end{sagesilent}

\begin{equation}
	\sage{bgmlv2_with_q}
\end{equation}


\noindent
This can be rearranged to express a bound on $d$ as follows
(recall from condition \ref{item:rankpos:lem:num_test_prob1}
in lemma \ref{lem:num_test_prob1} or corollary
\ref{cor:num_test_prob2} that $r>0$):


\begin{sagesilent}
from plots_and_expressions import bgmlv2_d_ineq
\end{sagesilent}
\begin{equation}
	\label{eqn-bgmlv2_d_upperbound}
	\sage{bgmlv2_d_ineq}
\end{equation}

\begin{sagesilent}
from plots_and_expressions import bgmlv2_d_upperbound_terms
\end{sagesilent}
Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
of $r$ again, there is a constant term
$\sage{bgmlv2_d_upperbound_terms.const}$,
a linear term
$\sage{bgmlv2_d_upperbound_terms.linear}$,
and a hyperbolic term
$\sage{bgmlv2_d_upperbound_terms.hyperbolic}$.
Notice that in the context of problem \ref{problem:problem-statement-2}
($\beta = \beta_{-}$),
the constant and linear terms match up with the ones
for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.

\subsubsection{
	Semistability of the Quotient:
	\texorpdfstring{
		$\Delta(v-u) \geq 0$
	}{
		Δ(v-u) ≥ 0
	}
}
\label{subsect-d-bound-bgmlv3}

This condition refers to condition
\ref{item:bgmlvv-u:lem:num_test_prob1}
from lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).

Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
$d$ yields:


\begin{sagesilent}
from plots_and_expressions import bgmlv3_d_upperbound_terms
\end{sagesilent}

\bgroup
\def\psi{\chern_1^{\beta}(v)}
\def\phi{\chern_2^{\beta}(v)}
\begin{equation*}
	\label{eqn-bgmlv3_d_upperbound}
	d \leq
	\sage{bgmlv3_d_upperbound_terms.linear}
	+ \sage{bgmlv3_d_upperbound_terms.const}
	+ \sage{bgmlv3_d_upperbound_terms.hyperbolic}
	\qquad
	\text{where }r>R
\end{equation*}
\egroup


\noindent
For $r=R$, $\Delta(v-u)\geq 0$ is always true, and for $r<R$ it gives a lower
bound on $d$, but it is weaker than the one given by the lower bound in
subsubsection \ref{subsect-d-bound-radiuscond}.
Viewing the right hand side of equation \ref{eqn-bgmlv3_d_upperbound}
as a function of $r$, the linear and constant terms almost match up with the
ones in the previous section, up to the 
$\chern_2^{\beta}(v)$ term.


However, when specializing to problem \ref{problem:problem-statement-2} again
(with $\beta = \beta_{-}$), then we have $\chern^{\beta}_2(v) = 0$.
And so in this context, the linear and constant terms do match up with the
previous subsubsections.

\subsubsection{All Bounds on $d$ Together for Problem
\ref{problem:problem-statement-2}}
\label{subsubsect:all-bounds-on-d-prob2}
%% RECAP ON INEQUALITIES TOGETHER

%%%% RATIONAL BETA MINUS
As mentioned in passing, when specializing to solutions $u$ of problem
\ref{problem:problem-statement-2}, the bounds on
$d=\chern^{\beta_{-}}_2(u)$ induced by conditions
\ref{item:bgmlvu:lem:num_test_prob2},
\ref{item:bgmlvv-u:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob1}
from corollary \ref{cor:num_test_prob2} have the same constant and linear
terms in $r$, but different hyperbolic terms.
These give bounds with the same assymptotes when we take $r\to\infty$
(for any fixed $q=\chern_1^{\beta_{-}}(u)$).

% redefine \beta (especially coming from rendered SageMath expressions)
% to be \beta_{-} for the rest of this subsubsection
\bgroup

\let\originalbeta\beta
\renewcommand\beta{{\originalbeta_{-}}}

\bgroup
% redefine \psi in sage expressions (placeholder for ch_1^\beta(F)
\def\psi{\chern_1^{\beta}(F)}
\begin{align}
	d &>&
	\frac{1}{2}\beta^2 r
	&+ \beta q,
	\phantom{+}& % to keep terms aligned
	 &\qquad\text{when\:} r > 0
	\label{eqn:radiuscond_d_bound_betamin}
\\
	d &\leq&
	\sage{bgmlv2_d_upperbound_terms.problem2.linear}
	&+ \sage{bgmlv2_d_upperbound_terms.problem2.const}
	+& \sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
	 &\qquad\text{when\:} r > 0
	 \label{eqn:bgmlv2_d_bound_betamin}
\\
	d &\leq&
	\sage{bgmlv3_d_upperbound_terms.problem2.linear}
	&+ \sage{bgmlv3_d_upperbound_terms.problem2.const}
	% ^ ch_2^\beta(F)=0 for beta_{-}
	+& \sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
	 &\qquad\text{when\:} r > R
	 \label{eqn:bgmlv3_d_bound_betamin}
\end{align}
\egroup


\begin{sagesilent}
from plots_and_expressions import \
bounds_on_d_qmin, \
bounds_on_d_qmax
\end{sagesilent}

\begin{figure}
\centering
\begin{subfigure}{.45\textwidth}
  \centering
	\sageplot[width=\linewidth]{bounds_on_d_qmin}
	\caption{$q = 0$ (all bounds other than green coincide on line)}
  \label{fig:d_bounds_xmpl_min_q}
\end{subfigure}%
\hfill
\begin{subfigure}{.45\textwidth}
  \centering
	\sageplot[width=\linewidth]{bounds_on_d_qmax}
	\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
  \label{fig:d_bounds_xmpl_max_q}
\end{subfigure}
\caption{
	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq\chern_0(E)$ for fixed, extreme,
	values of $q\coloneqq\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_extrm_q}
\end{figure}

Recalling that $q \coloneqq \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
it is worth noting that the extreme values of $q$ in this range lead to the
tightest bounds on $d$, as illustrated in figure
(\ref{fig:d_bounds_xmpl_extrm_q}).
In fact, in each case, one of the weak upper bounds coincides with one of the
weak lower bounds, (implying no possible destabilizers $E$ with
$\chern_0(E)=\vcentcolon r>R\coloneqq\chern_0(F)$ for these $q$-values).
This indeed happens in general since the right hand sides of
(eqn \ref{eqn:bgmlv2_d_bound_betamin}) and
(eqn \ref{eqn:radiuscond_d_bound_betamin}) match when $q=0$.
In the other case, $q=\chern^{\beta}_1(F)$, it is the right hand sides of
(eqn \ref{eqn:bgmlv3_d_bound_betamin}) and
(eqn \ref{eqn:radiuscond_d_bound_betamin}) which match.


The more generic case, when $0 < q\coloneqq\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
for the bounds on $d$ in terms of $r$ is illustrated in figure
(\ref{fig:d_bounds_xmpl_gnrc_q}).
The question of whether there are pseudo-destabilizers of arbitrarily large
rank, in the context of the graph, comes down to whether there are points
$(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (with large $r$)
% TODO have a proper definition for pseudo-destabilizers/walls
that fit above the yellow line (ensuring positive radius of wall) but below the
blue and green (ensuring $\Delta(u), \Delta(v-u) > 0$).
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:radiuscond_d_bound_betamin}).
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{sagesilent}
from plots_and_expressions import typical_bounds_on_d
\end{sagesilent}

\begin{figure}
\centering
\sageplot[width=\linewidth]{typical_bounds_on_d}
\caption{
	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
	value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_gnrc_q}
\end{figure}

\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{r} in Problem
\ref{problem:problem-statement-2}}

Now, the inequalities from the above subsubsection
\ref{subsubsect:all-bounds-on-d-prob2} will be used to find, for
each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave
no possible solutions for $d$. At that point, there are no solutions
$u=(r,c\ell,d\ell^2)$ to problem \ref{problem:problem-statement-2}.
The strategy here is similar to what was shown in theorem
\ref{thm:loose-bound-on-r}.


\renewcommand{\aa}{{a_v}}
\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\coloneqq \chern_1^{\beta}(E)
	  =\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} so that we are
considering $u$ which satisfy \ref{item:chern1bound:lem:num_test_prob2}
in corollary \ref{cor:num_test_prob2}.

Substituting the current values of $q$ and $\beta$ into the condition for the
radius of the pseudo-wall being positive
(eqn \ref{eqn:radiuscond_d_bound_betamin}) we get:

\begin{sagesilent}
from plots_and_expressions import \
positive_radius_condition_with_q, \
q_value_expr, \
beta_value_expr
\end{sagesilent}
\begin{equation}
\label{eqn:positive_rad_condition_in_terms_of_q_beta}
	\frac{1}{\lcm(m,2)}\ZZ
	\ni
	\qquad
	\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
	\qquad
	\in
	\frac{1}{2n^2}\ZZ
\end{equation}


\begin{sagesilent}
from plots_and_expressions import main_theorem1
\end{sagesilent}
\begin{theorem}[Bound on $r$ \#1]
\label{thm:rmax_with_uniform_eps}
	Let $v = (R,C\ell,D\ell^2)$ be a fixed Chern character. Then the ranks of the
	pseudo-semistabilizers for $v$,
	which are solutions to problem \ref{problem:problem-statement-2},
	with $\chern_1^\beta = q$
	are bounded above by the following expression.

	\bgroup
	\def\psi{\chern_1^{\beta}(F)}
	\renewcommand\Omega{{\lcm(m,2n^2)}}
	\begin{align*}
		\min
		\left(
			\sage{main_theorem1.r_upper_bound1}, \:\:
			\sage{main_theorem1.r_upper_bound2}
		\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 all solutions to problem
	\ref{problem:problem-statement-2}.
\end{theorem}

\begin{proof}

\noindent
Both $d$ and the lower bound in
(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
So, if any of the two upper bounds on $d$ come to within
$\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
$d$.
Hence any corresponding $r$ cannot be a rank of a
pseudo-semistabilizer for $v$.

To avoid this, we must have,
considering equations
\ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:radiuscond_d_bound_betamin}.

\bgroup

\let\originalepsilon\epsilon
\renewcommand\epsilon{{\originalepsilon_{v}}}

\begin{sagesilent}
from plots_and_expressions import \
assymptote_gap_condition1, assymptote_gap_condition2, kappa
\end{sagesilent}


\bgroup
\def\psi{\chern_1^{\beta}(F)}
\renewcommand\Omega{{\lcm(m,2n^2)}}
\begin{align}
	&\sage{assymptote_gap_condition1.subs(kappa==1)} \\
	&\sage{assymptote_gap_condition2.subs(kappa==1)}
\end{align}
\egroup

\noindent
This is equivalent to:

\bgroup
\renewcommand\Omega{{\lcm(m,2n^2)}}
\def\psi{\chern_1^{\beta}(F)}
\begin{equation}
	\label{eqn:thm-bound-for-r-impossible-cond-for-r}
	r \leq
	\min\left(
		\sage{
			main_theorem1.r_upper_bound1
		} ,
		\sage{
			main_theorem1.r_upper_bound2
		}
	\right)
\end{equation}
\egroup

\egroup % end scope where epsilon redefined

\end{proof}


\begin{sagesilent}
from plots_and_expressions import q_sol, Delta, psi
\end{sagesilent}

\begin{corollary}[Bound on $r$ \#2]
\label{cor:direct_rmax_with_uniform_eps}
	Let $v$ be a fixed Chern character and
	$R\coloneqq\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
	Then the ranks of the pseudo-semistabilizers for $v$,
	which are solutions to problem \ref{problem:problem-statement-2},
	are bounded above by the following expression.

	\bgroup
	\let\originalDelta\Delta
	\renewcommand\Delta{{\originalDelta(v)}}
	\renewcommand\Omega{{\lcm(m,2n^2)}}
	\begin{equation*}
		\sage{main_theorem1.corollary_r_bound}
	\end{equation*}
	\egroup
\end{corollary}

\begin{proof}
\bgroup
\renewcommand\Omega{{\lcm(m,2n^2)}}
\def\psi{\chern_1^{\beta}(F)}
\let\originalDelta\Delta
The ranks of the pseudo-semistabilizers for $v$ are bounded above by the
maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in theorem
\ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$:
\begin{equation*}
	f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
	f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
\end{equation*}
These have their minimums at $q=0$ and $q=\chern_1^{\beta}(F)$ respectively.
It suffices to find their intersection in
$q\in [0, \chern_1^{\beta}(F)]$, if it exists,
and evaluating on of the $f_i$ there.
The intersection exists, provided that
$f_1(\chern_1^{\beta}(F)) \geq f_2(\chern_1^{\beta}(F))=R$,
or equivalently,
$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
Solving for $f_1(q)=f_2(q)$ yields
\begin{equation*}
	q=\sage{q_sol.expand()}
\end{equation*}
And evaluating $f_1$ at this $q$-value gives:
\begin{equation*}
	\sage{main_theorem1.corollary_intermediate}
\end{equation*}
Finally, noting that $\originalDelta(v)=\psi^2\ell^2$, we get the bound as
stated in the corollary.
\egroup

\end{proof}

\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-second}
Just like in example \ref{exmpl:recurring-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$.

Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
$\sage{recurring.corrolary_bound} \approx  \sage{float(recurring.corrolary_bound)}$,
which is much closer to real maximum 25 than the original bound 144.
\end{example}

\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-second}
Just like in example \ref{exmpl:extravagant-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$.

Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
$\sage{extravagant.corrolary_bound} \approx  \sage{float(extravagant.corrolary_bound)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
\end{example}
%% refinements using specific values of q and beta

These bound can be refined a bit more by considering restrictions from the
possible values that $r$ take.
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.

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
$\aa,\bb$ specifically. The upcoming theorem (TODO ref) is less useful as a
`clean' formula for a bound on the ranks of the pseudo-semistabilizers, but has a
purpose in the context of writing a computer program to find
pseudo-semistabilizers. Such a program would iterate through possible values of
$q$, then iterate through values of $r$ within the bounds (dependent on $q$),
which would then determine $c$, and then find the corresponding possible values
for $d$.


Firstly, we only need to consider $r$-values for which $c\coloneqq\chern_1(E)$ is
integral:

\begin{equation}
	c =
	\sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
	\in \ZZ
\end{equation}

\noindent
That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
$n$, and so invertible mod $n$).


\noindent
Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.

Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_F$ in the
proof of theorem \ref{thm:rmax_with_uniform_eps}:

\begin{lemmadfn}[
	Finding a better alternative to $\epsilon_v$:
	$\epsilon_{v,q}$
	]
	\label{lemdfn:epsilon_q}
	Suppose $d \in \frac{1}{\lcm(m,2n^2)}\ZZ$ satisfies the condition in
	eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
	That is:

	\begin{equation*}
		\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
	\end{equation*}

	\noindent
	Then we have:

	\begin{equation}
		\label{eqn:epsilon_q_lemma_prop}
		d - \frac{(\aa r + 2\bb)\aa}{2n^2}
		\geq \epsilon_{v,q} \geq \epsilon_v > 0
	\end{equation}

	\noindent
	Where $\epsilon_{v,q}$ is defined as follows:

	\begin{equation*}
		\epsilon_{v,q} \coloneqq
		\frac{k_{q}}{\lcm(m,2n^2)}
	\end{equation*}
	with $k_{v,q}$ being the least $k\in\ZZ_{>0}$ satisfying
	\begin{equation*}
		k \equiv -\aa\bb \frac{m}{\gcd(m,2n^2)}
		\mod{\gcd\left(
			\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
			\frac{mn\aa}{\gcd(m,2n^2)}
		\right)}
	\end{equation*}
	
\end{lemmadfn}

\vspace{10pt}

\begin{proof}

Consider the following sequence of logical implications.
The one-way implication follows from
$\aa r + \bb \equiv 0 \pmod{n}$,
and the final logical equivalence is just a simplification of the expressions.

\begin{align}
	\frac{ x }{ \lcm(m,2) }
	- \frac{
		(\aa r+2\bb)\aa
	}{
		2n^2
	}
	= \frac{ k }{ \lcm(m,2n^2) }
	\quad \text{for some } x \in \ZZ
	\span \span \span \span \span
	\label{eqn:finding_better_eps_problem}
\\ \nonumber
\\ \Leftrightarrow& &
	- (\aa r+2\bb)\aa
	\frac{\lcm(m,2n^2)}{2n^2}
	&\equiv k &&
	\nonumber
\\ &&&
	\mod \frac{\lcm(m,2n^2)}{\lcm(m,2)}
	\span \span \span
	\nonumber
\\ \Rightarrow& &
	- \bb\aa
	\frac{\lcm(m,2n^2)}{2n^2}
	&\equiv k &&
	\nonumber
\\ &&&
	\mod \gcd\left(
		\frac{\lcm(m,2n^2)}{\lcm(m,2)},
		\frac{n \aa \lcm(m,2n^2)}{2n^2}
	\right)
	\span \span \span
	\nonumber
\\ \Leftrightarrow& &
	- \bb\aa
	\frac{m}{\gcd(m,2n^2)}
	&\equiv k &&
	\label{eqn:better_eps_problem_k_mod_n}
\\ &&&
	\mod \gcd\left(
		\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
		\frac{mn \aa}{\gcd(m,2n^2)}
	\right)
	\span \span \span
	\nonumber
\end{align}

In our situation, we want to find the least $k>0$ satisfying 
eqn \ref{eqn:finding_better_eps_problem}.
Since such a $k$ must also satisfy eqn \ref{eqn:better_eps_problem_k_mod_n},
we can pick the smallest $k_{q,v} \in \ZZ_{>0}$ which satisfies this new condition
(a computation only depending on $q$ and $\beta$, but not $r$).
We are then guaranteed that $k_{v,q}$ is less than any $k$ satisfying eqn
\ref{eqn:finding_better_eps_problem}, giving the first inequality in eqn
\ref{eqn:epsilon_q_lemma_prop}.
Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality:
$\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.

\end{proof}

\begin{sagesilent}
from plots_and_expressions import main_theorem2
\end{sagesilent}
\begin{theorem}[Bound on $r$ \#3]
\label{thm:rmax_with_eps1}
	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
	rational and expressed in lowest terms.
	Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with,
	which are solutions to problem \ref{problem:problem-statement-2},
	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
	are bounded above by the following expression:

	\bgroup
	\def\kappa{k_{v,q}}
	\def\psi{\chern_1^{\beta}(F)}
	\renewcommand\Omega{{\lcm(m,2n^2)}}
	\begin{align*}
		\min
		\left(
			\sage{main_theorem2.r_upper_bound1}, \:\:
			\sage{main_theorem2.r_upper_bound2}
		\right)
	\end{align*}
	\egroup
	Where $k_{v,q}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
	and $R = \chern_0(v)$

	Furthermore, if $\aa \not= 0$ then
	$r \equiv \aa^{-1}b_q \pmod{n}$.
\end{theorem}

Although the general form of this bound is quite complicated, it does simplify a
lot when $m$ is small.

\begin{sagesilent}
from plots_and_expressions import main_theorem2_corollary
\end{sagesilent}
\begin{corollary}[Bound on $r$ \#3 on $\PP^2$ and Principally polarized abelian surfaces]
\label{cor:rmax_with_eps1}
	Suppose we are working over $\PP^2$ or a principally polarized abelian surface
	(or any other surfaces with $m=1$ or $2$).
	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
	rational and expressed in lowest terms.
	Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with,
	which are solutions to problem \ref{problem:problem-statement-2},
	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
	are bounded above by the following expression:

	\bgroup
	\def\kappa{k_{v,q}}
	\def\psi{\chern_1^{\beta}(F)}
	\begin{align*}
		\min
		\left(
			\sage{main_theorem2_corollary.r_upper_bound1}, \:\:
			\sage{main_theorem2_corollary.r_upper_bound2}
		\right)
	\end{align*}
	\egroup
	Where $R = \chern_0(v)$ and $k_{v,q}$ is the least
	$k\in\ZZ_{>0}$ satisfying
	\begin{equation*}
		k \equiv -\aa\bb
		\pmod{n}
	\end{equation*}

	\noindent
	Furthermore, if $\aa \not= 0$ then
	$r \equiv \aa^{-1}b_q \pmod{n}$.
\end{corollary}

\begin{proof}
This is a specialisation of theorem \ref{thm:rmax_with_eps1}, where we can
drastically simplify the $\lcm$ and $\gcd$ terms by noting that $m$ divides both
$2$ and $2n^2$, and that $a_v$ is coprime to $n$.
\end{proof}

\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-third}
Just like in examples \ref{exmpl:recurring-first} and
\ref{exmpl:recurring-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{recurring.betaminus}$, giving $n=\sage{recurring.n}$
and $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
%% TODO transcode notebook code
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:

\begin{sagesilent}
from examples import bound_comparisons
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
\end{sagesilent}

\vspace{1em}
\noindent
\directlua{ table_width = 3*4+1 }
\begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
	$q=\chern_1^\beta(u)$
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	\\ \hline
	Thm \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	\\
	Thm \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
\end{tabular}
\vspace{1em}

\noindent
It's worth noting that the bounds given by theorem \ref{thm:rmax_with_eps1}
reach, but do not exceed the actual maximum rank 25 of the
pseudo-semistabilizers of $v$ in this case.
As a reminder, the original loose bound from theorem \ref{thm:loose-bound-on-r}
was 144.

\end{example}

\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-third}
Just like in examples \ref{exmpl:extravagant-first} and
\ref{exmpl:extravagant-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{extravagant.betaminus}$, giving $n=\sage{n:=extravagant.n}$
and $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
This example was chosen because the $n$ value is moderatly large, giving more
possible values for $k_{v,q}$, in dfn/lemma \ref{lemdfn:epsilon_q}. This allows
for a larger possible difference between the bounds given by theorems
\ref{thm:rmax_with_uniform_eps} and \ref{thm:rmax_with_eps1}, with the bound
from the second being up to $\sage{n}$ times smaller, for any given $q$ value.
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
in terms of the first few smallest possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:

\begin{sagesilent}
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(extravagant)
\end{sagesilent}


\vspace{1em}
\noindent
\directlua{ table_width = 12 }
\begin{tabular}{l\directlua{for i=0,table_width do tex.sprint([[|c]]) end}}
	$q=\chern_1^\beta(u)$
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
	\\ \hline
	Thm \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
	\\
	Thm \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
\end{tabular}
\vspace{1em}


\noindent
However the reduction in the overall bound on $r$ is not as drastic, since all
possible values for $k_{v,q}$ in $\{1,2,\ldots,\sage{n}\}$ are iterated through
cyclically as we consider successive possible values for $q$.
And for each $q$ where $k_{v,q}=1$, both theorems give the same bound.
Calculating the maximums over all values of $q$ yields
$\sage{max(theorem2_bounds)}$ for theorem \ref{thm:rmax_with_uniform_eps}, and
$\sage{max(theorem3_bounds)}$ for theorem \ref{thm:rmax_with_eps1}.
\end{example}

\egroup % end scope where beta redefined to beta_{-}

\subsubsection{All Semistabilizers Giving Sufficiently Large Circular Walls Left
of Vertical Wall}


Goals:
\begin{itemize}
	\item refresher on strategy
	\item point out no need for rational beta
	\item calculate intersection of bounds?
\end{itemize}

\subsection{Irrational $\beta_{-}$}

Goals:
\begin{itemize}
	\item Point out if only looking for sufficiently large wall, look at above
		subsubsection
	\item Relate to Pell's equation through coordinate change?
	\item Relate to numerical condition described by Yanagida/Yoshioka
\end{itemize}

\section{Computing solutions to Problem \ref{problem:problem-statement-2}}
\label{sect:prob2-algorithm}

Alongside this article, there is a library \cite{NaylorRust2023} to compute
the solutions to problem \ref{problem:problem-statement-2}, using the theorems
above.

The way it works, is by yielding solutions to the problem
$u=(r,c\ell,\frac{e}{2}\ell^2)$ as follows.

\subsection{Iterating Over Possible $q=\chern^{\beta_{-}}(u)$}

Given a Chern character $v$, the domain of the problem are first verified: that
$v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
$\beta_{-}(v)$ is rational.

Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
Iterate over $q \in [0,\chern_1^{\beta_{-}}(v)]\cap\frac{1}{n}\ZZ$.

For any $u = (r,c\ell,\frac{e}{2}\ell^2)$, satisfying
$\chern_1^{\beta_{-}}(u)=q$ for one of the $q$ considered is equivalent to
satisfying condition \ref{item:chern1bound:lem:num_test_prob2}
in corollary \ref{cor:num_test_prob2}.

\subsection{Iterating Over Possible $r=\chern_0(u)$ for Fixed $q=\chern^{\beta_{-}}(u)$}

Let $q=\frac{b_q}{n}$ be one of the values of $\chern_1^{\beta_{-}}(u)$ that we
have fixed. As mentioned before, the only values of $r$ which can
give $\chern_1^{\beta_{-}}(u)=q$ are precisely the ones which satisfy
$a_v r \equiv b_q \pmod{n}$.
This is true for all integers when $\beta_{-}=0$ (and so $n=1$), but otherwise,
this is equivalent to
$r \equiv {a_v}^{-1}b_q \pmod{n}$, since $a_v$ and $n$ are coprime.

Note that expressing $\mu(u)$ in term of $q$ and $r$ gives:
\begin{align*}
	\mu(u) & = \frac{c}{r} = \frac{q+r\beta_{-}}{r}
	\\
	&= \beta_{-} + \frac{q}{r}
\end{align*}

So condition \ref{item:mubound:lem:num_test_prob2} in corollary
\ref{cor:num_test_prob2} is satisfied at this point precisely when:

\begin{equation*}
	r > \frac{q}{\mu(u) - \beta_{-}}
\end{equation*}

Note that the right hand-side is greater than, or equal, to 0, so such $r$ also
satisfies \ref{item:rankpos:lem:num_test_prob2}.

Then theorem \ref{thm:rmax_with_eps1} gives an upper on possible $r$ values
for which it is possible to satisfy conditions
\ref{item:bgmlvu:lem:num_test_prob2},
\ref{item:bgmlvv-u:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob2}.

Iterate over such $r$ so that we are guarenteed to satisfy conditions
\ref{item:mubound:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob2}
in corollary
\ref{cor:num_test_prob2}, and have a chance at satisfying the rest.

\subsection{Iterating Over Possible $d=\chern_2(u)$ for Fixed $r=\chern_0(u)$
and $q=\chern^{\beta_{-}}(u)$}

At this point we have fixed $\chern_0(u)=r$ and
$\chern_1(u)=c=q+r\beta_{-}$.
And the cases considered are precisely the ones which satisfy conditions
\ref{item:chern1bound:lem:num_test_prob2},
\ref{item:mubound:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob2}
in corollary \ref{cor:num_test_prob2}.

It remains to find $\chern_2(u)=d=\frac{e}{2}$
which satisfy the remaining conditions
\ref{item:bgmlvu:lem:num_test_prob2},
\ref{item:bgmlvv-u:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob2}.
These conditions induce upper and lower bounds on $d$, and it then remains to
just pick the integers $e$ that give $d$ values within the bounds.

Thus, through this process yielding all solutions $u=(r,c\ell,\frac{e}{2}\ell^2)$
to the problem for this choice of $v$.



\newpage
\printbibliography

\newpage
\section{Appendix - SageMath code}

\usemintedstyle{tango}

\begin{footnotesize}
\inputminted[
	obeytabs=true,
	tabsize=2,
	breaklines=true,
	breakbefore=./
]{python}{filtered_sage.txt}
\end{footnotesize}

\end{document}