# I.variety() : missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex')

sage: f1=x1+x2+x3+x4+x5

sage: f2=x1*x2+x2*x3+x3*x4+x4*x5+x1*x5

sage: f3=x1*x2*x3+x2*x3*x4+x3*x4*x5+x4*x5*x1+x5*x1*x2

sage: f4=x1*x2*x3*x4+x2*x3*x4*x5+x3*x4*x5*x1+x4*x5*x1*x2+x5*x1*x2*x3

sage: f5=x1*x2*x3*x4*x5-1

sage: I = Ideal(f1,f2,f3,f4,f5)

sage: I.variety()


produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.

[{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]


which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

If I change the order from lex to degrevlex, I.variety() fails with the following error message:

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-479-cb6d916bc8b3> in <module>()
----> 1 I.variety()

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in __call__(self, *args, **kwds)
603         if not R.base_ring().is_field():
604             raise ValueError("Coefficient ring must be a field for function '%s'."%(self.f.__name__))
--> 605         return self.f(self._instance, *args, **kwds)
606
607 require_field = RequireField

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in variety(self, ring)
2667           if self.ring().term_order().is_global():
2668             verbose("Warning: falling back to very slow toy implementation.", level=0)
-> 2669             T = toy_variety.triangular_factorization(self.groebner_basis())
2670           else:
2671             raise TypeError("Local/unknown orderings not supported by 'toy_buchberger' implementation.")

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/toy_variety.pyc in triangular_factorization(B, n)
279   # recursively build the family,
280   # looping through the factors of p
--> 281   for (q,a) in p.factor():
282     # Construct an analog to I in (R.quotient(R.ideal(q)))[x_0,x_1,...x_{n-1}]
283     I = R.ideal([each.reduce([q]) for each in G])

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_element.pyc in factor(self, proof)
1662                 raise NotImplementedError("Factorization of multivariate polynomials over prime fields with characteristic > 2^29 is not implemented.")
1663         if proof:
-> 1664             raise NotImplementedError("proof = True factorization not implemented.  Call factor with proof=False.")
1665
1666         R._singular_().set_ring()


which is kind've mystifying, since we're working over either RR or CC, both of which have characteristic 0, and neither of which is a prime field. I thought maybe this is related to I.variety()'s odd behavior under lex order somehow, though, so it'd be worth including here.

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and ...

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