document Polyhedron defined over number field

[mbell]

As reported in this google group (​https://groups.google.com/forum/#!topic/sage-support/ew0bnGzjm98), it was not possible to create Polyhedron defined over number fields. Now that #17830 is merged it does work and it should be documented and even advertised in the documentation!

---

from the previous report:

To create polyhedra quickly, the final suggestion in the Polyhedron documentation (http://www.sagemath.org/doc/reference/geometry/sage/geometry/polyhedron/constructor.html#base-rings) is to work in a set number field. Although this appears to work for setting vertices, it does not appear to work for lines (or rays).

For example:

sage: var('x')
x
sage: K.<sqrt3> = NumberField(x^2-3, embedding=1.732)
sage: P = Polyhedron(lines=[(1,sqrt3)])
sage: P
A 1-dimensional polyhedron in (Number Field in sqrt3 with defining polynomial x^10 + x^2 - 3)^2 defined as the convex hull of 1 vertex and 2 rays
sage: P.rays()
(A ray in the direction (0, -sqrt3), A ray in the direction (1, sqrt3))

This should be compared with:

sage: P = Polyhedron(lines=[(1, sqrt(3))])
sage: P
A 1-dimensional polyhedron in (Symbolic Ring)^2 defined as the convex hull of 1 vertex and 1 line

and

sage: P = Polyhedron(lines=[(1, sqrt(3))], base_ring=AA)
sage: P
A 1-dimensional polyhedron in AA^2 defined as the convex hull of 1 vertex and 1 line

Additionally, how can a "1-dimensional polyhedron" be "defined as the convex hull of 1 vertex and 2 rays"?

As pointed out below this is an issue with Polyhedron using a number fields < comparison

sage: K.<x> = NumberField(x^3 - 1001, embedding=10)
sage: x > x + 1
True

[vbraun]

With 6.4.beta6 I get

sage: Polyhedron(lines=[(1,sqrt3)])
A 1-dimensional polyhedron in (Number Field in sqrt3 with defining polynomial x^2 - 3)^2 defined as the convex hull of 1 vertex and 1 line

[vbraun]

Though the original example fails:

sage: K.<L> = NumberField(x^3 + 3*x^2 - 83*x - 1022, embedding=11.6515)
sage: Polyhedron(eqns=[[0, -L - 2, -8, 2], [0, -2, -L + 6, 9], [0, -9, 5, -L - 7]])
A 3-dimensional polyhedron in (Number Field in L with defining polynomial x^3 + 3*x^2 - 83*x - 1022)^3 defined as the convex hull of 1 vertex, 2 rays, 1 line

Slightly simpler failure:

sage: Polyhedron(eqns=[[0, -L - 2, -8], [0, -2, -L + 6], [0, -9, 5]])
A 2-dimensional polyhedron in (Number Field in L with defining polynomial x^3 + 3*x^2 - 83*x - 1022)^2 defined as the convex hull of 1 vertex and 2 rays

[vbraun]

This originates at the following comparison:

sage: v = -1/392*L^2 - 45/392*L - 239/392
sage: v > 0
True
sage: v.n()
-2.29356924372991

[mbell]

Thanks for tracking this down. Should I file a separate ticket for this under the number theory?

[vbraun]

You can just change the description and use this ticket. I think we agree that this is not a bug in polyhedra. See also the discussion on sage-devel.

[vdelecroix]

Comparison is fixed for quadratic fields since #13213. To have it working on higher degree number field you could review #17830.

Vincent

[mkoeppe]

All of these tests in description and comments seem to working in 7.3.beta9.

[vdelecroix]

yes because comparison in number fields is now working... (#17830). Instead of making it invalid, it would be much better to add doctests!

[jipilab]

Since the Polyhedron evolved quite a bit in the last year, here is an update on Sage8.2.rc1:

sage: var('x')
x
sage: K.<sqrt3> = NumberField(x^2-3, embedding=1.732)
sage: P = Polyhedron(lines=[(1,sqrt3)])
sage: P.rays()
()
sage: P.lines()
(A line in the direction (1, sqrt3),)

sage: P = Polyhedron(lines=[(1, sqrt(3))])
Traceback (most recent call last):
...
ValueError: for polyhedra with floating point numbers, the only allowed ring is RDF with backend 'cdd'

sage: P = Polyhedron(lines=[(1, sqrt(3))], base_ring=AA)
sage: P
A 1-dimensional polyhedron in AA^2 defined as the convex hull of 1 vertex and 1 line

The value error above will be changed in #24835 to a more appropriate error message, for which:

sage: P = Polyhedron(lines=[(1, sqrt(3))],backend='cdd',base_ring=RDF)
sage: P
A 1-dimensional polyhedron in RDF^2 defined as the convex hull of 1 vertex and 1 line

is the only way to get a symbolic expression to pass on to the constructor of Polyhedron.

Concerning documenting the Polyhedron class usage through NumberFields?, I believe that #22572 does the job.

Hence, once #22572 is merged, I would set this ticket to won't fix: the bug is fixed.

[dimpase]

An improvement of the doc is posted on #26077, where I also point out that it is not sufficient to have a tutorial doc: the docstring of Polyhedron should have a real example with NF entries.

[jipilab]

Just an update of the current situation in 8.9beta3:

sage: var('x')
x
sage: K.<sqrt3> = NumberField(x^2-3, embedding=1.732)
sage: P = Polyhedron(lines=[(1,sqrt3)])
sage: P.rays()
()
sage: P.lines()
(A line in the direction (1, sqrt3),)
sage: P = Polyhedron(lines=[(1, sqrt(3))])
Traceback (most recent call last):
...
ValueError: no default backend for computations with Symbolic Ring
sage: P = Polyhedron(lines=[(1, sqrt(3))], base_ring=AA); P
A 1-dimensional polyhedron in AA^2 defined as the convex hull of 1 vertex and 1 line
sage: Q = Polyhedron(lines=[(1, sqrt(3))], backend='normaliz'); Q
A 1-dimensional polyhedron in (Symbolic Ring)^2 defined as the convex hull of 1 vertex and 1 line

and it is still possible to do:

sage: R = Polyhedron(lines=[(1, sqrt(3))],backend='cdd',base_ring=RDF);R
A 1-dimensional polyhedron in RDF^2 defined as the convex hull of 1 vertex and 1 line

Similar examples should now complete the docstring, now that #25097 is merged.