Opened 8 years ago

# document Polyhedron defined over number field

Reported by: Owned by: mbell major sage-7.4 number theory Polyhedron, Number field chapoton, jipilab, jakobkroeker N/A #21105

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
```

### comment:1 Changed 8 years ago by 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
```

### comment:2 Changed 8 years ago by 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
```
Last edited 8 years ago by vbraun (previous) (diff)

### comment:3 follow-up: ↓ 4 Changed 8 years ago by 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
```

### comment:4 in reply to: ↑ 3 Changed 8 years ago by mbell

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

### comment:5 Changed 8 years ago by 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.

### comment:6 Changed 8 years ago by mbell

• Component changed from geometry to number theory
• Description modified (diff)

### comment:7 Changed 7 years ago by vdelecroix

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

Vincent

### comment:8 Changed 6 years ago by mkoeppe

• Milestone changed from sage-6.4 to sage-duplicate/invalid/wontfix
• Status changed from new to needs_review

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

### comment:9 Changed 6 years ago by vdelecroix

• Milestone changed from sage-duplicate/invalid/wontfix to sage-7.3
• Status changed from needs_review to needs_work

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

### comment:10 Changed 6 years ago by vdelecroix

• Description modified (diff)
• Summary changed from Bug creating Polyhedron from lines in number field to document Polyhedron defined over number field

### comment:11 Changed 6 years ago by mkoeppe

• Dependencies set to #21105

### comment:12 Changed 6 years ago by mkoeppe

• Milestone changed from sage-7.3 to sage-7.4

### comment:15 Changed 4 years ago by 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.

### comment:16 Changed 4 years ago by 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.

### comment:17 Changed 3 years ago by 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.

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