Creating homomorphisms of relative number fields seems totally broken

[williamstein]

The following _should_ create the correct homomorphism (complex conjugation, see #4725):

sage: K.<j,b> = QQ[sqrt(-1), sqrt(2)]
sage: conj = K.hom([-j, b])
boom!

However it doesn't.

CC: @lftabera

Component: number fields

Author: Robert Harron

Reviewer: Francis Clarke

Issue created by migration from https://trac.sagemath.org/ticket/4726

[rharron]

Author: Robert Harron

[rharron]

comment:3

I just uploaded a patch that allows you to construct relative number field morphisms as above (the code simply wasn't written accept something like that). I just made it go recursively through the list you provide, so in a tower of fields, you only need to provide the images up to the point where your morphism is the identity.

I'll set this to needs review after I do a full doctest overnight (the number_field module passes all doctests).

[sagetrac-fwclarke]

comment:5

This mostly works well. I tried it on some cases where the relative
degrees were different. For example

sage: C.<z> = CyclotomicField(15)
sage: K = C.relativize(z^5 + 1, 'a'); K
Number Field in a0 with defining polynomial x^4 - a1*x^3 + (a1 - 1)*x^2 + x - a1 over its base field
sage: K.inject_variables()
Defining a0, a1
sage: L = C.relativize(z^4 + z, 'b'); L
Number Field in b0 with defining polynomial x^2 - b1*x - 1/2*b1^3 + b1^2 - b1 - 1/2 over its base field
sage: H = Hom(K, L)
sage: K.hom(map(H[2], K.gens()))
Relative number field morphism:
  From: Number Field in a0 with defining polynomial x^4 - a1*x^3 + (a1 - 1)*x^2 + x - a1 over its base field
  To:   Number Field in b0 with defining polynomial x^2 - b1*x - 1/2*b1^3 + b1^2 - b1 - 1/2 over its base field
  Defn: a0 |--> -b0 + b1
        a1 |--> -1/2*b1^3 + b1^2 - b1 + 1/2

My concerns are about when the default base homomorphism is used. It is certainly not correct to describe default_base_hom as "trivial", and not clear anyway what that would mean if the domain and codomain differ. Its docstring says

Pick an embedding of the base field of self into the codomain of this homset. This is done in an essentially arbitrary way.

Since the value is cached, see line 526 of sage/rings/number_field/morphism.py (after your patch is applied), using the default argument base_hom=None should always give the same restriction to the base_field. However, continuing the above example,

sage: [K.hom([h(a0)])(a1) for h in H]
[-1/2*b1^3 + b1^2 - b1 + 1/2,
 1/2*b1^3 - b1^2 + b1 + 1/2,
 -1/2*b1^3 + b1^2 - b1 + 1/2,
 1/2*b1^3 - b1^2 + b1 + 1/2,
 -1/2*b1^3 + b1^2 - b1 + 1/2,
 1/2*b1^3 - b1^2 + b1 + 1/2,
 1/2*b1^3 - b1^2 + b1 + 1/2,
 -1/2*b1^3 + b1^2 - b1 + 1/2]

This happens because
sage.rings.number_field.morphism.RelativeNumberFieldHomset._from_im doesn't do what it claims. With again the same definitions:

sage: b0 = L.gen(); b1 = L.base_field().gen()
sage: base_hom = K.base_field().hom([1/2*b1^3 - b1^2 + b1 + 1/2]); base_hom
Ring morphism:
  From: Number Field in a1 with defining polynomial x^2 - x + 1
  To:   Number Field in b1 with defining polynomial x^4 - x^3 + 2*x^2 + x + 1
  Defn: a1 |--> 1/2*b1^3 - b1^2 + b1 + 1/2
sage: H._from_im([b0], base_hom)
Relative number field morphism:
  From: Number Field in a0 with defining polynomial x^4 - a1*x^3 + (a1 - 1)*x^2 + x - a1 over its base field
  To:   Number Field in b0 with defining polynomial x^2 - b1*x - 1/2*b1^3 + b1^2 - b1 - 1/2 over its base field
  Defn: a0 |--> b0
        a1 |--> -1/2*b1^3 + b1^2 - b1 + 1/2

which does not restrict to the base_field correctly. I note that at present _from_im is not used anywhere else in Sage.

In fact since

sage: K.absolute_generator()
a0

there is only one homomorphism from K to L that sends a0 to b0, so an error should have been raised by H._from_im([b0], base_hom).

[rharron]

comment:6

Nice catch. I've written a fix for _from_im and have mostly done writing up code that given a "short" list for im_gens attempts to find a base_hom that works. I'll upload it when I get that done.

[rharron]

comment:9

Attachment: trac_4726_enhance_relative_number_morphism_constructor.patch.gz

Alright, I've updated the patch. I added a check in _from_im for whether the morphism agrees with base_hom. And I added a new method _from_im_without_base_hom which is like _from_im, but attempts to find a base_hom compatible with the im_gen passed.

[sagetrac-fwclarke]

comment:10

This improves things significantly.

But I'm not too happy about _from_im having the default check=False. For all other constructors of homomorphisms, the check paramater has default True. The idea being that other methods can set it as False in cases where checking has already been done; see #10843, which still needs reviewing (once it's rebased).

The real problem is that the present code for _from_im is mathematically incorrect. Once a rigourous version is defined, it is possible to write _from_im_without_base_hom much more simply.

I attach a patch (to be applied after your latest patch) which implements these changes. In addition it rewrites default_base_hom, the point being that it is unnecessary to cache its output since caching is already done by embeddings.

There are some problems too with the docstring for RelativeNumberFieldHomset.__call__. In particular, it is not true that "if the list specifies a morphism that maps the generators of the base fields to themselves, then truncating that list will yield the same morphism." For example,

sage: K.<a,b,c> = NumberField([x^2 - 2, x^2 - 3, x^2 - 5])
sage: K.hom([-a, b, -c])
Relative number field endomorphism of Number Field in a with defining polynomial x^2 - 2 over its base field
  Defn: a |--> -a
        b |--> b
        c |--> -c
sage: K.hom([-a])
Relative number field endomorphism of Number Field in a with defining polynomial x^2 - 2 over its base field
  Defn: a |--> -a
        b |--> b
        c |--> c

I also think that there needs to be a warning that a homomorphism for which a partial list of generators is given may not be uniquely defined. In
other words, the "essentially arbitrary" remarks in the docstrings for _from_im_without_base_hom and default_base_hom need promoting to this level. Moreover the resulting modification to the syntax of hom for relative number fields needs to be directly available to users. At the moment K.hom?, where K is a relative number field, yields the docstring for the generic sage.structure.parent_gens.ParentWithGens.hom.

[sagetrac-fwclarke]

Reviewer: Francis Clarke

[sagetrac-fwclarke]

Attachment: trac_4726_reviewer.patch.gz

To be applied on top of trac_4726_enhance_relative_number_morphism_constructor.patch