Outline the new algorithm

main.tex

@@ -745,6 +745,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 		\item $\Delta(v-u) \geq 0$
 			\label{item:bgmlvv-u:lem:num_test_prob2}
 		\item $\mu(u)=\frac{c}{r}<\mu(v)$
+			\label{item:mubound:lem:num_test_prob2}
 		\item $0\leq\chern_1^{\beta_{-}}(u)\leq\chern_1^{\beta_{-}}(v)$
 			\label{item:chern1bound:lem:num_test_prob2}
 		\item $\chern_2^{\beta_{-}}(u)>0$
@@ -2105,6 +2106,93 @@ Goals:
 	\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