main.tex

for Fixed
\texorpdfstring{$r=\chern_0(u)$}{r}
and
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
}

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 $v$.



\onlyifstandalone{
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}

\end{document}
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