Algebraic polyhedra with Normaliz / e-antic

[mkoeppe]

Implements polyhedra over embedded algebraic number fields.

• Preliminary setup within sage: install e-antic, upgrade normaliz/pynormaliz (see upgrade normaliz to 3.7.1, pynormaliz to 2.1 and package e-antic #27682)

• the input is allowed to come from various fields, which will be coerced to a number field first via number_field_elements_from_algebraics should create embedded number field elements #20181:

        sage: x = polygen(ZZ); P = Polyhedron(vertices=[[sqrt(2)], [AA.polynomial_root(x^3-2, RIF(0,3))]], backend='normaliz')

• See doctests in src/sage/geometry/polyhedron/backend_normaliz.py and src/sage/geometry/polyhedron/library.py

• For debugging help, we can write out Normaliz input files and PyNormaliz function calls as a side effect by passing verbose=True.

Depends on #25091
Depends on #27731
Depends on #27965

CC: @w-bruns @videlec @jplab @sebasguts @sagetrac-tmonteil @mo271

Component: geometry

Keywords: IMA-PolyGeom

Author: Matthias Koeppe

Branch/Commit: f94bf37

Reviewer: Vincent Delecroix, Jean-Philippe Labbé

Issue created by migration from https://trac.sagemath.org/ticket/25097

[videlec]

comment:1

prerequisite: package antic (that has no release yet).

[mkoeppe]

Branch: u/mkoeppe/qnormaliz-algebraic

[mkoeppe]

Changed branch from u/mkoeppe/qnormaliz-algebraic to public/25097/qnormaliz-algebraic

[mkoeppe]

Dependencies: #25090

[mkoeppe]

Last 10 new commits:

117e428	Upgrade normaliz to 3.5.3
7057ef4	Merge branch to get docstring adaptation
b6ae686	Merge branch 'develop' of git://trac.sagemath.org/sage into t/25090/pynormaliz2
eaedcc2	Update PyNormaliz to 1.14
4dbdc7d	Updated the Cone calls format
1dcef4c	_init_from_normaliz_data: New, use it in _init_from_*representation
fcd8101	Updated pynormaliz to 1.15
19d8539	Polyhedron_normaliz: In verbose mode, write out normaliz format files
4cfcd14	_normaliz_format: Fix last change
ea7a6b8	Polyhedron_normaliz: Prepare classes for work over field extensions

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1 +1,25 @@
+As a first step, we make it possible to write Normaliz output files when setting up a polyhedron.
 
+```
+sage: P = Polyhedron(vertices=polytopes.icosahedron().vertices(), backend='normaliz', verbose='out.in')
+# Calling PyNormaliz.NmzCone(**{'subspace': [], 'vertices': [[0, 2, sqrt5 + 1, 4], [0, -2, sqrt5 + 1, 4], [2, sqrt5 + 1, 0, 4], [2, -sqrt5 - 1, 0, 4], [sqrt5 + 1, 0, 2, 4], [sqrt5 + 1, 0, -2, 4], [-2, sqrt5 + 1, 0, 4], [-2, -sqrt5 - 1, 0, 4], [-sqrt5 - 1, 0, 2, 4], [0, 2, -sqrt5 - 1, 4], [0, -2, -sqrt5 - 1, 4], [-sqrt5 - 1, 0, -2, 4]], 'cone': []})
+# Wrote equivalent Normaliz input file to out.in
+sage: !cat out.in
+amb_space 3
+number_field min_poly (a^2 - 5) embedding [2.236067977499789 +/- 8.06e-16]
+subspace 0
+vertices 12
+ 0 2 (a + 1) 4
+ 0 -2 (a + 1) 4
+ 2 (a + 1) 0 4
+ 2 (-a - 1) 0 4
+ (a + 1) 0 2 4
+ (a + 1) 0 -2 4
+ -2 (a + 1) 0 4
+ -2 (-a - 1) 0 4
+ (-a - 1) 0 2 4
+ 0 2 (-a - 1) 4
+ 0 -2 (-a - 1) 4
+ (-a - 1) 0 -2 4
+cone 0
+```

[mkoeppe]

Commit: ea7a6b8

[mkoeppe]

Description changed:

--- 
+++ 
@@ -23,3 +23,9 @@
  (-a - 1) 0 -2 4
 cone 0
 ```
+
+Next steps to be able to generate more examples:
+- Fix up `_init_from_Hrepresentation` so it does not try to cancel.
+- #20181 (`number_field_elements_from_algebraics` should create embedded number field elements) so that we can have more polyhedra over numberfields (instead of `AA`...)
+
+

[mkoeppe]

comment:8

Could someone who has worked on / has opinions on the Polyhedron constructor code help here?

What should we do for polyhedra over AA such as the following:

sage: P = polytopes.regular_polygon(3)
sage: P.base_ring()
Algebraic Real Field

If backend=normaliz is requested, should we cast to the appropriate number field via #20181? Or should it be an error?

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,4 +1,4 @@
-As a first step, we make it possible to write Normaliz output files when setting up a polyhedron.
+As a first step, we make it possible to write Normaliz output files while setting up a polyhedron over a numberfield. (#25090 added writing Normaliz output files in the rational case.)
 
 ```
 sage: P = Polyhedron(vertices=polytopes.icosahedron().vertices(), backend='normaliz', verbose='out.in')

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,3 +1,10 @@
+**Normaliz side of things:**
+- clone from https://github.com/mkoeppe/Normaliz/tree/enfnormaliz2018 
+- run script `install_normaliz_with_opt.sh` to install FLINT, ARB, ANTIC (unreleased), E-ANTIC (unreleased)
+- then can test with `Qsource/Qnormaliz Qtest/test-nf/dodecahedron-v.in && cat Qtest/test-nf/dodecahedron-v.out`
+
+**Sage side of things:**
+
 As a first step, we make it possible to write Normaliz output files while setting up a polyhedron over a numberfield. (#25090 added writing Normaliz output files in the rational case.)
 
 ```

[mkoeppe]

Description changed:

--- 
+++ 
@@ -31,6 +31,8 @@
 cone 0
 ```
 
+(Most examples in `Qtest/test-nf/` have been generated like this.)
+
 Next steps to be able to generate more examples:
 - Fix up `_init_from_Hrepresentation` so it does not try to cancel.
 - #20181 (`number_field_elements_from_algebraics` should create embedded number field elements) so that we can have more polyhedra over numberfields (instead of `AA`...)

[mkoeppe]

Description changed:

--- 
+++ 
@@ -37,4 +37,8 @@
 - Fix up `_init_from_Hrepresentation` so it does not try to cancel.
 - #20181 (`number_field_elements_from_algebraics` should create embedded number field elements) so that we can have more polyhedra over numberfields (instead of `AA`...)
 
+Other next steps:
+- merge with Vincent's master branch of e-antic
+- (Result of discussion of mkoeppe, gh-sebasguts): `PyQNormaliz`... use strings to transport arbs and lists for everything else. We do not use `sage.libs.arb` etc. so we can keep PyQNormaliz standalone.
+- (Result of discussion of mkoeppe, jipilab:) `Polyhedron_normaliz` should accept `AA`, `CyclotomicField`, etc. input and use these as a `base_ring`. When preparing data for `PyQNormaliz`, use `number_field_elements_from_algebraics`. When reading data from `PyQNormaliz`, first build number field elements, then inject into the correct `base_ring`.
 

[jplab]

Changed dependencies from #25090 to #25090, #20181

[jplab]

comment:14

What should we do for polyhedra over AA such as the following:

sage: P = polytopes.regular_polygon(3)
sage: P.base_ring()
Algebraic Real Field

If backend=normaliz is requested, should we cast to the appropriate number field via #20181? Or should it be an error?

I would say the cleanest thing to do once #20181 is done will be to use it! This probably will also simplify the constructor for field.