VIII. What do you get from seeing equations?: Variables separated
equations encodes Galois theory into 
algebraic equations.<br>
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VIII.1. Three standout degrees 7, 13, 15: Cassao-Nogu&egrave;s and
Couveignes produced the
actual equations of the families arising in #b [CoCa99].
That was 30 years after I wrote my paper. Further, they did not find
another route to what was in my paper. Rather, they used my approach to
the group theory of the example. It is even harder to figure
what comes from having the equations, rather than just knowing from the
Branch Cycle Lemma (and braid action on Nielsen classes) the nature of
the families of covers. <br>
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VIII.2. Siegel's Theorem and separated variable equations: That brings
up the significance
of the Avanzi-Zannier, Bilu-Tichy, et. al. papers that treated the
variables separated subject (find the genus 0 and 1 factors of
variables separated equations). They tried to write out the equations.
I haven't figured the significance of looking at the coefficients,
except it seems to feel right to most people that seeing the equations
is "good." <br>
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<a name="VIII.3._The_generalization_of_MacCluers"></a>VIII.3. The
generalization of MacCluer's Theorem: <br>
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VIII.4. Final remarks on Variables separated equations: What they often
miss in
looking at the equations is the nature and significance of getting the
right equivalence relation on covers so the parameter space of the
covers would yield valuable information. Without this you cannot do
such things as my identification of the rational function description
of exceptional covers with Serre's Open Image Theorem. Without this you
would not come up with this addition to Peter Mueller's description of
geometric monodromy groups of polynomials, which is in my Thompson
Birthday paper. You see from Mueller's classification that all the
families of genus 0 covers that arise, the only that have non-trivial
families came up in the solution of Davenport's Problem. For each of
the degrees 7, 15, 21 none of those families have definition field Q,
and there are several families of each (known precisely from the Branch
Cycle Lemma and the braid orbits). Now the punch line. If you take the
vector bundle attached to each cover, then -- for each of those degrees
-- there is really just one connected family of reduced equivalence
classes of covers. Further, the parameter space for each family is a
genus 0 curve. More so, those three genus 0 curves -- upper half plane
quotients covering the j-line, branched over \infty and the two
elliptic points -- though they are not modular curves, stand out for
very practical problems as an analog of finite collection of genus 0
modular curves. That can't be the end of the story, because the
significance of the genus 0 modular curves ended up in Monstrous
Moonshine. I haven't yet -- maybe I won't even bother, for that
requires an idea no one can count on easily -- made the analog case.
What I do claim, however, is that the Davenport problem analog opens up
the territory of modular curve thinking to many more mathematicians and
the hard work – often little recognized – they do.&nbsp; Add the start
of this story
to the variables separated html file.
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