number_field_elements_from_algebraics should create embedded number field elements

[mkoeppe]

This is a follow-up on #17830 and #18819.

#17830 made this possible:

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

But if one uses number_field_elements_from_algebraics, one gets only non-embedded number field elements:

sage: x = polygen(ZZ)
sage: K, cbrt2, hom = number_field_elements_from_algebraics(AA.polynomial_root(x^3-2, RIF(0,3)))
sage: 6064/4813 < cbrt2 < 90325/71691
False            # (an perhaps an error in Python 3?)

This ticket adds an option embedded to create embedded number field elements.

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

TODECIDE:

• Close ticket QQbar should have a coercion from number fields with embedding #5355?

CC: @vbraun @gagern @videlec @sagetrac-tmonteil @jjermann @staroste @jplab @mo271 @yuan-zhou @JohnCremona @tscrim

Component: number fields

Keywords: IMA-PolyGeom

Author: Jean-Philippe Labbé

Branch: 9255f2c

Reviewer: Matthias Koeppe, John Cremona

Issue created by migration from https://trac.sagemath.org/ticket/20181

[mkoeppe]

comment:1

While looking at related tickets, I noticed the very old ticket #5355, which should be closed I think.

[mkoeppe]

Description changed:

--- 
+++ 
@@ -12,7 +12,8 @@
 But if one uses `number_field_elements_from_algebraics`, one gets only non-embedded number field elements:
 
 ```
-sage: cbrt2 = number_field_elements_from_algebraics(AA.polynomial_root(x^3-2, RIF(0,3)))
+sage: x = polygen(ZZ)
+sage: K, cbrt2, hom = number_field_elements_from_algebraics(AA.polynomial_root(x^3-2, RIF(0,3)))
 sage: 6064/4813 < cbrt2 < 90325/71691
 False
 ```

[jplab]

Commit: 26dc3e4

[jplab]

Branch: public/algnum_to_numberfield

[jplab]

comment:5

Here is a first version. Comments are welcome.

One question is whether RealIntervalField is the right choice?

---

New commits:

26dc3e4

Added option for embedded NumberField

[jplab]

comment:6

Maybe it would be nice to also add the optional parameter to the method as_number_field_element as well...

[jplab]

comment:7

The following should work but it does not:

sage: UCF = UniversalCyclotomicField()
sage: E = UCF.gen(5)
sage: my_nums = [-52*E - 136*E^2 - 136*E^3 - 52*E^4, -66*E - 168*E^2 - 168*E^3 - 66*E^4, -46*E - 114*E^2 - 114*E^3 - 46*E^4, -24*E - 58*E^2 - 58*E
....: ^3 - 24*E^4, sqrt(2)]
sage: [n(_) for _ in my_nums]
[187.914855054991 - 8.67361737988404e-18*I,
 231.039466852489 - 1.04083408558608e-17*I,
 156.026311234993 - 8.67361737988404e-18*I,
 79.0131556174964 - 4.33680868994202e-18*I,
 1.41421356237310]
sage: res = number_field_elements_from_algebraics(my_nums,embedded=True)
Traceback (most recent call last):
...
TypeError: unable to convert -0.4370160244488211? - 1.344997023927915?*I to real interval

The choice of root shoould always be made real...

[sagetrac-git]

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

db778ee

Made it possible to deal with complex number fields

[sagetrac-git]

Changed commit from 26dc3e4 to db778ee

[jplab]

comment:9

It is now possible to get this:

sage: UCF = UniversalCyclotomicField()
sage: E = UCF.gen(5)
sage: L.<b> = NumberField(x^2-189*x+16, embedding=200)
sage: my_nums = [-52*E - 136*E^2 - 136*E^3 - 52*E^4, L.gen()._algebraic_(AA),sqrt(2)]
sage: aa_my_nums = [AA(_) for _ in my_nums]
sage: res = number_field_elements_from_algebraics(aa_my_nums,embedded=True)
sage: res
(Number Field in a with defining polynomial y^8 - 35670*y^6 + 476899047*y^4 - 2832410271650*y^2 + 6305298701739921,
 [2310/26212773509*a^7 - 185432947/78638320527*a^5 + 1652517502195/78638320527*a^3 - 4904676315215467/78638320527*a + 94,
  -1238/2803377488467023*a^7 + 185460719/11213509953868092*a^5 - 2754936849443/11213509953868092*a^3 + 8180694680816975/3737836651289364*a + 189/2,
  -1979/1887160880826*a^7 + 26472586/943580440413*a^5 - 235822245043/943580440413*a^3 + 466325019915415/629053626942*a],
 Ring morphism:
   From: Number Field in a with defining polynomial y^8 - 35670*y^6 + 476899047*y^4 - 2832410271650*y^2 + 6305298701739921
   To:   Algebraic Real Field
   Defn: a |--> 96.9475535136628?)
sage: res[0].gen_embedding()
96.9475535136628?
sage: res = number_field_elements_from_algebraics(aa_my_nums,embedded=False)
sage: res[0].gen_embedding()  # Nothing is returned(!?!?!)
sage:

Perhaps it would be nice to say that there is no embedding instead of returning nothing?

[jplab]

Author: Jean-Philippe Labbé

[mkoeppe]

comment:12

The returned morphism is not able to convert the elements.

sage: res[2](my_nums[0])
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-8-74370e91d982> in <module>()
----> 1 res[Integer(2)](my_nums[Integer(0)])

/Users/mkoeppe/s/sage/some-sage-install-prefix/lib/python2.7/site-packages/sage/categories/map.pyx in sage.categories.map.Map.__call__ (build/cythonized/sage/categories/map.c:6978)()
    781                 x = D(x)
    782             except (TypeError, NotImplementedError):
--> 783                 raise TypeError("%s fails to convert into the map's domain %s, but a `pushforward` method is not properly implemented" % (x, D))
    784         else:
    785             x = converter(x)

TypeError: -52*E(5) - 136*E(5)^2 - 136*E(5)^3 - 52*E(5)^4 fails to convert into the map's domain Number Field in a with defining polynomial y^8 - 35670*y^6 + 476899047*y^4 - 2832410271650*y^2 + 6305298701739921, but a `pushforward` method is not properly implemented

Edit: I withdraw this comment; I misremembered what hom is supposed to do.