OK, I'll finish this later but just report:
solve has domain flag (needs docstring and test)
Sometimes a polynomial can be solved over the RR domain even though a
general solution cannot be given. To get this behavior, use the domain
flag, e.g.
>>> var('y', real=True)
y
>>> eq = S('''81*x**7/25 - 693*x**6/25 + 1875611*x**5/15750 -
>>> 125971061*x**4/378000
... + 529468867*x**3/810000 - 16336527047*x**2/17010000 +
... 79077332921*x/68040000 - 2307430519/1890000 + 597749683/(567000*x) -
... 1035948691/(1512000*x**2) + 880580863/(2721600*x**3) -
... 75961409/(680400*x**4) + 1061551/(38880*x**5) - 1355/(324*x**6) +
... 25/(81*x**7)''').subs(x, y)
...
>>> solve(eq)
[]
>>> solve(eq, domain='RR')
[0.493925305792679, 0.686078618485917]
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