main.tex

vertical wall (TODO as discussed in ref).

% 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 &\geq&
	\sage{bgmlv1_d_lowerbound_linear_term}
	&+ \sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
	+& \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)},
	 &\qquad\text{when\:} r > \frac{R}{2}
	 \label{eqn:bgmlv1_d_bound_betamin}
\\
	d &\leq&
	\sage{bgmlv2_d_upperbound_linear_term}
	&+ \sage{bgmlv2_d_upperbound_const_term}
	+& \sage{bgmlv2_d_upperbound_exp_term},
	 &\qquad\text{when\:} r > 0
	 \label{eqn:bgmlv2_d_bound_betamin}
\\
	d &\leq&
	\sage{bgmlv3_d_upperbound_linear_term}
	&+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)}
	% ^ ch_2^\beta(F)=0 for beta_{-}
	+& \sage{bgmlv3_d_upperbound_exp_term_alt2},
	 &\qquad\text{when\:} r > R
	 \label{eqn:bgmlv3_d_bound_betamin}
\end{align}
\egroup

Furthermore, we get an extra bound for $d$ resulting from the condition that the
radius of the circular wall must be positive. As discussed in (TODO ref), this
is equivalent to $\chern^{\beta}_2(E) > 0$, which yields:

\begin{sagesilent}
positive_radius_condition = (
	(
		(0 > - u.twist(beta).ch[2])
		+ d # rearrange for d
	)
	.subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q
	.expand()
)
\end{sagesilent}

\begin{equation}
	\label{eqn:positive_rad_d_bound_betamin}
	\sage{positive_radius_condition}
\end{equation}
\begin{sagesilent}
def beta_min(chern):
  ts = stability.Tilt()
  return min(
    map(
      lambda soln: soln.rhs(),
      solve(
        (ts.degree(chern))
          .expand()
          .subs(ts.alpha == 0),
        beta
      )
    )
  )

v_example = Chern_Char(3,2,-2)
q_example = 7/3

def plot_d_bound(
  v_example,
  q_example,
  ymax=5,
  ymin=-2,
  xmax=20,
  aspect_ratio=None
):

  # Equations to plot imminently representing the bounds on d:
  eq1 = (
    bgmlv1_d_lowerbound
    .subs(R == v_example.ch[0])
    .subs(C == v_example.ch[1])
    .subs(D == v_example.ch[2])
    .subs(beta = beta_min(v_example))
    .subs(q == q_example)
  )

  eq2 = (
    bgmlv2_d_upperbound
    .subs(R == v_example.ch[0])
    .subs(C == v_example.ch[1])
    .subs(D == v_example.ch[2])
    .subs(beta = beta_min(v_example))
    .subs(q == q_example)
  )

  eq3 = (
    bgmlv3_d_upperbound
    .subs(R == v_example.ch[0])
    .subs(C == v_example.ch[1])
    .subs(D == v_example.ch[2])
    .subs(beta = beta_min(v_example))
    .subs(q == q_example)
  )

  eq4 = (
    positive_radius_condition.rhs()
    .subs(q == q_example)
    .subs(beta = beta_min(v_example))
  )

  example_bounds_on_d_plot = (
    plot(
      eq3,
      (r,v_example.ch[0],xmax),
      color='green',
			linestyle = "dashed",
      legend_label=r"upper bound: $\Delta(G) \geq 0$",
    )
    + plot(
      eq2,
      (r,0,xmax),
      color='blue',
			linestyle = "dashed",
      legend_label=r"upper bound: $\Delta(E) \geq 0$"
    )
    + plot(
      eq4,
      (r,0,xmax),
      color='orange',
			linestyle = "dotted",
      legend_label=r"lower bound: $\mathrm{ch}_2^{\beta_{-}}(E)>0$"
    )
    + plot(
      eq1,
      (r,v_example.ch[0]/2,xmax),
      color='red',
			linestyle = "dotted",
      legend_label=r"lower bound: $\Delta(E) + \Delta(G) \leq \Delta(F)$"
    )
  )
  example_bounds_on_d_plot.ymin(ymin)
  example_bounds_on_d_plot.ymax(ymax)
  example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])
  if aspect_ratio:
    example_bounds_on_d_plot.set_aspect_ratio(aspect_ratio)
  return example_bounds_on_d_plot

\end{sagesilent}

\begin{figure}
\centering
\begin{subfigure}{.45\textwidth}
  \centering
	\sageplot[width=\linewidth]{plot_d_bound(v_example, 0)}
	\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]{plot_d_bound(v_example, 4)}
	\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 $d:=\chern_2(E)$ in terms of $r:=\chern_0(E)$ for fixed, extreme,
	values of $q:=\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_extrm_q}
\end{figure}

Recalling that $q := \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
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
$\chern_0(E)=:r>R:=\chern_0(F)$ 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:positive_rad_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:positive_rad_d_bound_betamin}) which match.


The more generic case, when $0 < q:=\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
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
$(r,d) \in \ZZ \oplus \frac{1}{m} \ZZ$ (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 $\Delta(E), \Delta(G) > 0$).
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:positive_rad_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{figure}
\centering
\sageplot[
	width=\linewidth
]{plot_d_bound(v_example, 2, ymax=6, ymin=-0.5, aspect_ratio=1)}
\caption{
	Bounds on $d:=\chern_2(E)$ in terms of $r:=\chern_0(E)$ for a fixed
	value $\chern_1^{\beta}(F)/2$ of $q:=\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_gnrc_q}
\end{figure}

\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{r}}

Now, the inequalities from the above (TODO REF) 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 Chern characters
$(r,c,d)$ that satisfy all inequalities to give a pseudowall.


\subsubsection{All Semistabilizers Left of Vertical Wall for Rational Beta min}


The strategy here is similar to what was shown in (sect \ref{sec:twisted-chern}),
% ref to Schmidt?

\begin{sagesilent}
var("a_v b_q n") # Define symbols introduce for values of beta and q
beta_value_expr = (beta == a_v/n)
q_value_expr = (q == b_q/n)
\end{sagesilent}

\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:=\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}).

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

\begin{equation}
\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}

\begin{sagesilent}
var("nu", domain="real") # placeholder for the specific values of 1/epsilon

assymptote_gap_condition1 = (1/nu < bgmlv2_d_upperbound_exp_term)
assymptote_gap_condition2 = (1/nu < bgmlv3_d_upperbound_exp_term_alt2)

r_upper_bound1 = (
	assymptote_gap_condition1
	* r * nu
)

assert r_upper_bound1.lhs() == r

r_upper_bound2 = (
	assymptote_gap_condition2
	* (r-R) * nu + R
)

assert r_upper_bound2.lhs() == r
\end{sagesilent}

\begin{theorem}[Bound on $r$ \#1]
\label{thm:rmax_with_uniform_eps}
	Let $v = (R,C,D)$ be a fixed Chern character. Then the ranks of the
	pseudo-semistabilizers for $v$ with
	$\chern_1^\beta = q$
	are bounded above by the following expression.

	\bgroup
	\def\nu{\lcm(m,2n^2)}
	\def\psi{\chern_1^{\beta}(F)}
	\begin{align*}
		\min
		\left(
			\sage{r_upper_bound1.rhs()}, \:\:
			\sage{r_upper_bound2.rhs()}
		\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 pseudo-semistabilizers for $v$.
\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:positive_rad_d_bound_betamin}.

\bgroup

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

\begin{sagesilent}
var("epsilon")

# Tightness conditions:

bounds_too_tight_condition1 = (
	bgmlv2_d_upperbound_exp_term
	< epsilon
)

bounds_too_tight_condition2 = (
	bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)
	< epsilon
)
\end{sagesilent}

\bgroup
\def\psi{\chern_1^{\beta}(F)}
\begin{equation}
	\min\left(
		\sage{bgmlv2_d_upperbound_exp_term},
		\sage{bgmlv3_d_upperbound_exp_term_alt2}
	\right)
	\geq \epsilon := \frac{1}{\lcm(m,2n^2)}
\end{equation}
\egroup

\noindent
This is equivalent to:

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

\egroup % end scope where epsilon redefined

\end{proof}

\begin{sagesilent}
var("Delta", domain="real")
q_sol = solve(r_upper_bound1.rhs() == r_upper_bound2.rhs(), q)[0].rhs()
r_upper_bound_all_q = (
	r_upper_bound1.rhs()
	.expand()
	.subs(q==q_sol)
	.subs(psi**2 == Delta)
	.subs(1/psi**2 == 1/Delta)
)
\end{sagesilent}

\begin{corrolary}[Bound on $r$ \#2]
	\label{cor:direct_rmax_with_uniform_eps}
	Let $v$ be a fixed Chern character and
	$R:=\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2)\Delta(v)$.
	Then the ranks of the pseudo-semistabilizers for $v$
	are bounded above by the following expression.

	\bgroup
	\let\originalDelta\Delta
	\def\nu{\lcm(m,2n^2)}
	\renewcommand\Delta{{\originalDelta(v)}}
	\begin{equation*}
		\sage{r_upper_bound_all_q.expand()}
	\end{equation*}
	\egroup
\end{corrolary}

%% TODO simplified expression for rmax by predicting which q gives rmax

%% refinements using specific values of q and beta

This 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
$\frac{1}{m}\ZZ$ is at least $\frac{1}{\lcm(m,2n^2)}$ 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 $c:=\chern_1(E)$ 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 $n$).

\begin{sagesilent}
	rhs_numerator = (
	positive_radius_condition
	.rhs()
	.subs([q_value_expr,beta_value_expr])
	.factor()
	.numerator()
	)
\end{sagesilent}

\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 better alternatives to $\epsilon_F$:
	$\epsilon_{q,1}$ and $\epsilon_{q,2}$
	]
	\label{lemdfn:epsilon_q}
	Suppose $d \in \frac{1}{m}\ZZ$ satisfies the condition in
	eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
	That is:

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

	\noindent
	Then we have:

	\begin{equation*}
		d - \frac{(\aa r + 2\bb)\aa}{2n^2}
		\geq \epsilon_{q,2} \geq \epsilon_{q,1} > 0
\end{equation*}

Where $\epsilon_{q,1}$ and $\epsilon_{q,2}$ are defined as follows:

\begin{equation*}
	\epsilon_{q,1} :=
	\frac{k_{q,1}}{2mn^2}
	\qquad
	\epsilon_{q,2} :=
	\frac{k_{q,2}}{2mn^2}
\end{equation*}
\begin{align*}
	\text{where }
	&k_{q,1} \text{ is the least }
	k\in\ZZ_{>0}\: s.t.:\:
	k \equiv -\aa\bb m \mod n
\\
	&k_{q,2} \text{ is the least }
	k\in\ZZ_{>0}\: s.t.:\:
	k \equiv \aa\bb m (\aa\aa^{'}-2)
	\mod n\gcd(2n,\aa^2 m)
\end{align*}
	
\end{lemmadfn}

It is worth noting that $\epsilon_{q,2}$ is potentially larger than
$\epsilon_{q,2}$
but calculating it involves a $\gcd$, a modulo reduction, and a modulo $n$
inverse, for each $q$ considered.

\begin{proof}

Consider the following:

\begin{align}
	\frac{ x }{ m }
	- \frac{
		(\aa r+2\bb)\aa
	}{
		2n^2
	}
	= \frac{ k }{ 2mn^2 }
	\quad \text{for some } x \in \ZZ
	\span \span \span \span \span
	\label{eqn:finding_better_eps_problem}
\\ &\Longleftrightarrow& 
	- (\aa r+2\bb)\aa m
	&\equiv k &&
	\mod 2n^2
\\ &\Longleftrightarrow&
	- \aa^2 m r - 2\aa\bb m
	&\equiv k &&
	\mod 2n^2
\\ &\Longrightarrow&
  \aa^2 \aa^{'}\bb m - 2\aa\bb m
	&\equiv k &&
	\mod \gcd(2n^2, \aa^2 mn)
	\label{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
\\ &\Longrightarrow&
  -\aa\bb m
	&\equiv k &&
	\mod n
	\label{eqn:better_eps_problem_k_mod_n}
\end{align}

In our situation, we want to find the least $k$ 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 $k_{q,1} \in \ZZ_{>0}$ which satisfies this new condition
(a computation only depending on $q$ and $\beta$, but not $r$).
We are then guaranteed that the gap $\frac{k}{2mn^2}$ is at least
$\epsilon_{q,1}$.
Furthermore, $k$ also satisfies
eqn \ref{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
so we can also pick the smallest $k_{q,2} \in \ZZ_{>0}$ satisfying this condition,
which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_{q,2}$.

\end{proof}


\begin{theorem}[Bound on $r$ \#3]
\label{thm:rmax_with_eps1}
	Let $v$ be a fixed Chern character, with $\frac{a_F}{n}=\beta:=\beta(v)$
	rational and expressed in lowest terms.
	Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with
	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
	are bounded above by the following expression (with $i=1$ or $2$).

\begin{sagesilent}
var("delta", domain="real") # placeholder symbol to be replaced by k_{q,i}
eps_k_i_subs = nu == (2*m*n^2)/delta
\end{sagesilent}

	\bgroup
	\def\delta{k_{q,i}}
	\def\psi{\chern_1^{\beta}(F)}
	\begin{align*}
		\min
		\left(
			\sage{r_upper_bound1.rhs().subs(eps_k_i_subs)}, \:\:
			\sage{r_upper_bound2.rhs().subs(eps_k_i_subs)}
		\right)
	\end{align*}
	\egroup
	Where $k_{q,i}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
	and $R = \chern_0(v)$

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


\minorheading{Irrational $\beta$}

\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}

\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}