Newer
Older
\bgroup
\def\psi{\chern_1^{\beta}(F)}
\begin{equation}
\min\left(
\sage{bgmlv2_d_upperbound_exp_term},
\sage{bgmlv3_d_upperbound_exp_term_alt2}
\geq \epsilon := \frac{1}{\lcm(m,2n^2)}
\begin{sagesilent}
# rearrange the "tightness" conditions in terms of r
bounds_too_tight_condition1 = (
(bounds_too_tight_condition1 * r / epsilon)
.expand()
)
bounds_too_tight_condition2 = (
(bounds_too_tight_condition2 * (r - R) / epsilon + R)
.expand()
)
# Check that these are indeed rearranged for r
assert bounds_too_tight_condition1.rhs() == r
assert bounds_too_tight_condition2.rhs() == r
\end{sagesilent}
\noindent
This is equivalent to:
\bgroup
\def\psi{\chern_1^{\beta}(F)}
\def\Delta{\lcm(m,2n^2)}
\label{eqn:thm-bound-for-r-impossible-cond-for-r}
r \leq
\egroup % end scope where epsilon redefined
%% 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 $\epsilon_F$:
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:
\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 tautology:
- \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}
&\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}
\label{eqn:better_eps_problem_k_mod_n}
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 $\epsilon_{q,2}$.
\begin{theorem}[Bound on $r$ \#3]
\label{thm:rmax_with_eps1}
Let $v = (R,C,D)$ be a fixed Chern character. Then the ranks of the
pseudo-semistabilizers for $v$ with
$\chern_1^\beta = q = \frac{a_q}{n}$
are bounded above by the following expression (with $i=1$ or 2).
\begin{equation*}
\min
\left(
q^2,
2R\beta q
+C^2
-2DR
-2Cq
+q^2
+\frac{R}{\lcm(m,2n^2)}
Where $\epsilon_{q,i}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q}.
\minorheading{Irrational $\beta$}
\egroup % end scope where beta redefined to beta_{-}
\section{Appendix - SageMath code}
\usemintedstyle{tango}
\inputminted[
obeytabs=true,
tabsize=2,
breaklines=true,
breakbefore=./
]{python}{filtered_sage.txt}