symbolic placeholder for complex root

[rwst]

The Sage equivalent of SymPy's CRootOf (=ComplexRootOf) is missing, which is just a symbolic placeholder for a solution that cannot be displayed symbolically.

sage: from sympy import solve as ssolve
sage: ssolve(x^6+x+1, x)

[CRootOf(x**6 + x + 1, 0),
 CRootOf(x**6 + x + 1, 1),
 CRootOf(x**6 + x + 1, 2),
 CRootOf(x**6 + x + 1, 3),
 CRootOf(x**6 + x + 1, 4),
 CRootOf(x**6 + x + 1, 5)]
sage: (_[0]+1)._sage_()
...
AttributeError: 'ComplexRootOf' object has no attribute '_sage_'

Defect because conversion from SymPy fails.

A possible solution for #11941 depends on this.

Depends on #24062

Component: symbolics

Author: Ralf Stephan

Branch: ba171aa

Reviewer: Emmanuel Charpentier

Issue created by migration from https://trac.sagemath.org/ticket/22024

[nbruin]

comment:1

I think it's a bit more than a symbolic placeholder. It does seem there's an embedding in CC associated with it. So conversion to sage should probably be by converting to QQbar. Something like this might work:

QQbar.polynomial_root(ZZ['x'](c.poly.as_expr()._sage_()),CIF(c.n()._sage_()))

but there will be a lot of corner cases to deal with too.

[rwst]

comment:2

Replying to @nbruin:

I think it's a bit more than a symbolic placeholder. It does seem there's an embedding in CC associated with it. So conversion to sage should probably be by converting to QQbar. Something like this might work:

QQbar.polynomial_root(ZZ['x'](c.poly.as_expr()._sage_()),CIF(c.n()._sage_()))

but there will be a lot of corner cases to deal with too.

To return a QQbar element would certainly be ideal. However, we're dealing at the moment with an unnecessary error, and returning a list of objects that show the polynomial plus the float value (and from which both can be extracted) would be sufficient to fix that defect.

[rwst]

comment:3

To return a float with arbitrary precision we can use SymPy's n() as well, so we just keep the original SymPy object.

[rwst]

Branch: u/rws/symbolic_placeholder_for_complex_root

[rwst]

Commit: ba171aa

[rwst]

Author: Ralf Stephan

[rwst]

Dependencies: #24062

[EmmanuelCharpentier]

comment:6

Builds, passes ptestlong with no error whatsoever.

I'm a bit stimyed by the usefulness of the result. If this is supposed to just allow the conversion of a SymPy computation into Sage, it does the job :

sage: sage: from sympy import solve as ssolve
....: sage: foo=ssolve(x^6+x+1, x)
....: 
sage: [t._sage_() for t in foo]

[complex_root_of(x^6 + x + 1, 0),
 complex_root_of(x^6 + x + 1, 1),
 complex_root_of(x^6 + x + 1, 2),
 complex_root_of(x^6 + x + 1, 3),
 complex_root_of(x^6 + x + 1, 4),
 complex_root_of(x^6 + x + 1, 5)]
sage: [t._sage_().n() for t in foo]

[-0.790667188814418 - 0.300506920309552*I,
 -0.790667188814418 + 0.300506920309552*I,
 -0.154735144496843 - 1.03838075445846*I,
 -0.154735144496843 + 1.03838075445846*I,
 0.945402333311260 - 0.611836693781009*I,
 0.945402333311260 + 0.611836693781009*I]

bit, as Nil's remarked, I find that we loose the properties of an answer in QQbar. However, the guarantee oif a fixed order may be the key to further computations.

==>positive_review

But see my forthcoming comment to #11941 : we may have bigger fish to fry...