Add DomainError positive char univariate polynomial rings

[GiacomoPope]

Following the discussion in #62

[oscarbenjamin]

These should be something like NotImplementedError at least rather than AttributeError. The meaning of AttributeError is that an object does not have an attribute. It should not be used for anything else.

[oscarbenjamin]

This looks good.

We don't necessarily need to do it here but to complete these we should add an implementation of real_roots for arb_poly, fmpz_poly and fmpq_poly.

For arb_poly it is:

def real_roots(self):
    return self.roots()

For fmpz_poly we just need a square-free factorisation (all poly have this now after gh-189). Then convert to arb_poly to compute the roots:

def real_roots(self):
    _, fac = self.factor_squarefree()
    roots = []
    for f, m in fac:
        roots.extend([(r, m) for r, m in arb_poly(f).roots()])
    # Sort the roots?
    return roots

For fmpq_poly the real roots are the real roots of the numerator:

def real_roots(self):
    return self.numer().real_roots()

Oh, wait arb_poly does not have roots:

In [4]: arb_poly([1, 0, 1]).roots()
...
NotImplementedError: Polynomial has no factor method, roots cannot be determined

I guess we need to add real_roots to acb_poly using acb_poly_validate_real_roots and then arb_poly can have:

def roots(self):
    return acb_poly(self).real_roots()

Oh, and there is an interface inconsistency:

In [12]: acb_poly([1, 0, -1]).roots()
Out[12]: [-1.00000000000000, 1.00000000000000]

In [13]: fmpz_poly([1, 0, -1]).roots()
Out[13]: [(1, 1), (-1, 1)]

I suppose that acb_poly and arb_poly could have a different interface for now...

[oscarbenjamin]

Or maybe for fmpz_poly.real_roots it should just filter the roots from fmpz_poly.complex_roots(). There seems to be
arb_fmpz_poly_complex_roots but no separate method for real roots.

[oscarbenjamin]

I assume that there is some advantage in using arb_fmpz_poly_complex_roots rather than acb_poly_validate_real_roots for the case where we have exact fmpz coefficients. Not sure, but isolating all complex roots is a vastly more expensive operation than isolating only real roots...

[oscarbenjamin]

Okay, let's just merge this for now and consider proper real_roots support later.