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luke naylor latex documents
research
Max Destabilizer Rank
Commits
e4c7684c
Commit
e4c7684c
authored
1 year ago
by
Luke Naylor
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Adjust stronger theorem to more general expressions
parent
75e95026
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#28799
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1 year ago
Stage: test
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main.tex
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View file @
e4c7684c
...
...
@@ -1403,7 +1403,7 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
$
\epsilon
_{
v,q
}$
]
\label
{
lemdfn:epsilon
_
q
}
Suppose
$
d
\in
\frac
{
1
}{
2
}
\ZZ
$
satisfies the condition in
Suppose
$
d
\in
\frac
{
1
}{
\lcm
(
m,
2
n
^
2
)
}
\ZZ
$
satisfies the condition in
eqn
\ref
{
eqn:positive
_
rad
_
condition
_
in
_
terms
_
of
_
q
_
beta
}
.
That is:
...
...
@@ -1425,51 +1425,72 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
\begin{equation*}
\epsilon
_{
v,q
}
\coloneqq
\frac
{
k
_{
q
}}{
2n
^
2
}
\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*}
with
$
k
_{
v,q
}$
being the least
$
k
\in\ZZ
_{
>
0
}$
satisfying
$
k
\equiv
-
\aa\bb
\mod
n
$
\end{lemmadfn}
\vspace
{
10pt
}
\begin{proof}
Consider the following:
\begin{align}
\frac
{
x
}{
2
}
\frac
{
x
}{
\lcm
(m,2)
}
-
\frac
{
(
\aa
r+2
\bb
)
\aa
}{
2n
^
2
}
=
\frac
{
k
}{
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
}
\\
\Longleftrightarrow
&
&
- (
\aa
r+2
\bb
)
\aa
\frac
{
\lcm
(m,2n
^
2)
}{
2n
^
2
}
&
\equiv
k
&&
\mod
n
^
2
\\
\Longleftrightarrow
&
&
-
\aa
^
2 r - 2
\aa\bb
&
\equiv
k
&&
\mod
n
^
2
\\
\Longrightarrow
&
&
\aa
^
2
\aa
^{
-1
}
\bb
- 2
\aa\bb
\mod
\frac
{
\lcm
(m,2n
^
2)
}{
\lcm
(m,2)
}
\nonumber
\\
\Longrightarrow
&
&
-
\bb\aa
\frac
{
\lcm
(m,2n
^
2)
}{
2n
^
2
}
&
\equiv
k
&&
\mod
n
\label
{
eqn:better
_
eps
_
problem
_
k
_
mod
_
gcd2n2
_
a2mn
}
\nonumber
\\
&&&
\mod
\gcd\left
(
\frac
{
\lcm
(m,2n
^
2)
}{
\lcm
(m,2)
}
,
\frac
{
mn
\aa
}{
\gcd
(m,2n
^
2)
}
\right
)
\span
\span
\span
\nonumber
\\
\Longleftrightarrow
&
&
-
\aa\bb
-
\bb\aa
\frac
{
m
}{
\gcd
(m,2n
^
2)
}
&
\equiv
k
&&
\mod
n
\label
{
eqn:better
_
eps
_
problem
_
k
_
mod
_
n
}
\\
&&&
\mod
\gcd\left
(
\frac
{
n
^
2
\gcd
(m,n)
}{
\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
$
satisfying
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,
1
}
\in
\ZZ
_{
>
0
}$
which satisfies this new condition
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
...
...
@@ -1495,6 +1516,7 @@ from plots_and_expressions import r_upper_bound1, r_upper_bound2
\bgroup
\def\kappa
{
k
_{
v,q
}}
\def\psi
{
\chern
_
1
^{
\beta
}
(F)
}
\renewcommand\Omega
{{
\lcm
(m,2n
^
2)
}}
\begin{align*}
\min
\left
(
...
...
@@ -1507,7 +1529,7 @@ from plots_and_expressions import r_upper_bound1, r_upper_bound2
and
$
R
=
\chern
_
0
(
v
)
$
Furthermore, if
$
\aa
\not
=
0
$
then
$
r
\equiv
\aa
^{
-
1
}
b
_
q
(
\mod
n
)
$
.
$
r
\equiv
\aa
^{
-
1
}
b
_
q
\
p
mod
{
n
}
$
.
\end{theorem}
...
...
@@ -1571,7 +1593,7 @@ 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
}$
smaller, for any given
$
q
$
value.
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:
...
...
@@ -1612,6 +1634,7 @@ end}
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
(
theorem
2
_
bounds
)
}$
for theorem
\ref
{
thm:rmax
_
with
_
uniform
_
eps
}
, and
$
\sage
{
max
(
theorem
3
_
bounds
)
}$
for theorem
\ref
{
thm:rmax
_
with
_
eps1
}
.
...
...
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