Fix Conic doctests

[mstreng]

The following optional Magma doctest fails:

File "src/sage/schemes/plane_conics/con_field.py", line 1073, in sage.schemes.plane_conics.con_field.ProjectiveConic_field.rational_point
Failed example:
    Conic([L.gen(), 30, -21]).has_rational_point(algorithm='magma') # optional - magma
Expected:
    False
Got:
    True

This is due to the following bug in Conics over number fields: if the cache of a conic is empty and the user asks whether a rational point exists without asking for the point itself, then has_rational_point always returns True. Here is a non-Magma example documenting this bug:

sage: P.<a> = QuadraticField(2)                                                 
sage: C = Conic(P,[1,1,1])                                                      
sage: C.has_rational_point()                                                    
True
sage: C.has_rational_point(point=True)                                          
(False, None)
sage: C.has_rational_point()                                                    
True
sage: C.has_rational_point(obstruction=True)                                    
(True, None)
sage: C.has_rational_point(point=True, obstruction=True)                        
(False,
 Ring morphism:
   From: Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?
   To:   Algebraic Real Field
   Defn: a |--> -1.414213562373095?)
sage: C.has_rational_point()                                                    
False

Moreover, the documentation incorrectly claims (see #31892)

• that the inverse maps of conic parametrization are incorrect and
• that the is_onemethod is a good way to check this.

CC: @kliem @JohnCremona @fchapoton @tscrim

Component: geometry

Keywords: conic, magma

Author: Marco Streng

Branch/Commit: 8f12947

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/33603

[mstreng]

Branch: u/mstreng/fix_conic_doctests

[mstreng]

Changed keywords from none to conic, magma

[mstreng]

Author: Marco Streng

[mstreng]

New commits:

af11c80

Doctest fixes for Conics

[mstreng]

Commit: af11c80

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

714d21a	Instead of removing the incorrect cache check, copy the correct cache check from conics over QQ
b991973	Add non-Magma example that fails on [SageMath](../wiki/SageMath) 9.5

[sagetrac-git]

Changed commit from af11c80 to b991973

[mstreng]

comment:6

Apparently the cache bug was introduced in #28900.

[tscrim]

Reviewer: Travis Scrimshaw

[tscrim]

comment:7

I guess this comes from _finite_obstructions and _infinite_obstructions being None, which carries a different meaning. Could you add a comment saying something like, "These values might be None, so we explicitly check against a list"? Otherwise if doctests pass (I am assuming this is ready for review; if so, please set it to needs review), then positive review.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

8f12947

Add comment explaining why we check against a list.

[sagetrac-git]

Changed commit from b991973 to 8f12947

[mstreng]

comment:9

All (long) non-optional tests passed. All Magma tests in src/sage/schemes/plane_conics passed. And then I added the comment that Travis asked for.

[tscrim]

comment:10

Thank you. LGTM.

[vbraun]

Changed branch from u/mstreng/fix_conic_doctests to 8f12947