Comparison of number field elements dependent of real embedding

[vdelecroix]

Comparison of embedded number field elements should be identical to the comparison of the embedding. In other words, we should have:

sage: x = polygen(ZZ)
sage: K.<cbrt2> = NumberField(x^3 - 2, embedding=AA.polynomial_root(x^3-2, RIF(0,3)))
sage: 6064/4813 < cbrt2 < 90325/71691
True

This has been done in #13213 for quadratic number fields. This ticket aims to implement this for all number fields by storing into the parent a list of interval approximations of the generator.

This concerns only real embeddings.

To go further, one should consider:

• having a fast evaluation of integer polynomial with real intervals (mpfi elements) input
• make better interaction with QQbar (see also #18103 and #18104)

[vdelecroix]

Working implementation (i.e. all tests pass in sage.rings.number_fields). Needs some speedup and bullet proofs!

---

New commits:

​1fd916f	trac #17830: ordering for real embedded number fields
​316dd8e	trac #17830: floor/ceil
​5f7f57b	trac #17830: cleaner import in number_field.py

[sstarosta]

I have done some testing and committed a few tweaks done on Sage days 66 + doc improvement.

[vdelecroix]

I am not sure it is because of your commit, but

sage -t continued_fraction.py  # 1 doctest failed
sage -t polynomial/polynomial_rational_flint.pyx  # 1 doctest failed

[vdelecroix]

The two failing doctests come from the introduction of the method _real_mpfi_. I just propose to get rid of it in this ticket. The main reason is that there are a lot of duplication between:

• AA and QQbar
• RLF and CLF
• what is implemented in this ticket for number fields

I propose to clean it up in ticket #18103 and add the _real_mpfi_ feature there.

Vincent

---

New commits:

​d0f6f7b	Trac 17830: simpler initialization + TODO
​96ac61e	Trac 17830: revert the introduction of _real_mpfi_

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​90b4acc	trac #17830: ordering for real embedded number fields
​ee5a198	trac #17830: floor/ceil
​ec50d4d	trac #17830: cleaner import in number_field.py
​f07dc08	trac #17830: doc improvements and small tweaks
​b7ba83a	Trac 17830: simpler initialization + TODO

[vdelecroix]

rebase on sage-6.7.beta3

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​95f3ada	trac #17830: ordering for real embedded number fields
​7d0d0c0	trac #17830: floor/ceil
​2d49517	trac #17830: cleaner import in number_field.py
​e1efbd6	trac #17830: doc improvements and small tweaks
​875f756	Trac 17830: simpler initialization + TODO
​a8c294b	Trac #17830: rich_to_bool is no longer a method

[vdelecroix]

rebased on 6.8.beta6

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​575b264	Trac #18159: add a ._test_cardinality in Sets category
​cc87bd5	Trac #18159: remove a duplicated test
​32c403d	Trac #18159: fix cardinality/is_finite in sets/set.py
​937ebf4	Trac #18159: fix some tests related to TestSuite
​eb50b94	Trac #18159: fix two parents

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​95f3ada	trac #17830: ordering for real embedded number fields
​7d0d0c0	trac #17830: floor/ceil
​2d49517	trac #17830: cleaner import in number_field.py
​e1efbd6	trac #17830: doc improvements and small tweaks
​875f756	Trac 17830: simpler initialization + TODO
​a8c294b	Trac #17830: rich_to_bool is no longer a method

[vdelecroix]

Sorry... I pushed to the wrong ticket. Back to normal.

[chapoton]

+Decreased doctests rings/number_field/number_field_base.pyx from 11 / 11 = 100% to 12 / 13 = 92%

[vdelecroix]

Replying to chapoton:

+Decreased doctests rings/number_field/number_field_base.pyx from 11 / 11 = 100% to 12 / 13 = 92%

Then what? There is no way to test _init_embedding_approx that initialize private attributes. For sure we can duplicate doctests from _get_embedding_approx or even write

TESTS::

    sage: 1 + 1 # indirect doctest
    2

if it makes you happier. But I am not happy with that. If you have any constructive suggestion I would be happy to hear it.

General note: asking for review (= waiting from other people comments) is different from merging into Sage. Patchbot green light concerns the latter.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​5b7e757	trac #17830: ordering for real embedded number fields
​bd7eef0	trac #17830: floor/ceil
​94b22b6	trac #17830: cleaner import in number_field.py
​138957a	trac #17830: doc improvements and small tweaks
​9313849	Trac 17830: simpler initialization + TODO

[vdelecroix]

New commits:

​5b7e757	trac #17830: ordering for real embedded number fields
​bd7eef0	trac #17830: floor/ceil
​94b22b6	trac #17830: cleaner import in number_field.py
​138957a	trac #17830: doc improvements and small tweaks
​9313849	Trac 17830: simpler initialization + TODO

[git]

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

​c7a50e7

Trac #17830: add a useless doctest

[vbraun]

Lgtm but merge conflict...

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​ff13d1f	trac #17830: ordering for real embedded number fields
​c52229d	trac #17830: floor/ceil
​8543255	trac #17830: cleaner import in number_field.py
​c098dc7	trac #17830: doc improvements and small tweaks
​8bb1031	Trac #17830: simpler initialization + TODO
​d63b7ac	Trac #17830: add a useless doctest

[vdelecroix]

Thanks for having a look! Rebased on 7.0.beta1.

Vincent