Fix comparison of automorphisms of Eisenstein extensions

[pjbruin]

In SageMath 9.3.beta1:

sage: R.<x> = QQ[]
....: K = Qp(19)
....: L.<b> = K.extension(x^2 - 19)
....: a = L.hom([b], codomain=L)
....: id = L.Hom(L).identity()
....: a == id
Traceback (most recent call last):
...
TypeError: 'NoneType' object is not iterable
During handling of the above exception, another exception occurred:
...
AttributeError: 'NoneType' object has no attribute 'list'
During handling of the above exception, another exception occurred:
...
TypeError: cannot convert x to a p-adic element
During handling of the above exception, another exception occurred:
...
TypeError: None fails to convert into the map's domain 19-adic Eisenstein Extension Field in b defined by x^2 - 19, but a `pushforward` method is not properly implemented

CC: @jdemeyer @tscrim @roed314

Component: padics

Author: Peter Bruin

Branch/Commit: u/pbruin/30938-padic_element_from_None @ 64063cf

Issue created by migration from https://trac.sagemath.org/ticket/30938

[pjbruin]

Branch: u/pbruin/30938-padic_element_from_None

[pjbruin]

comment:1

The attached branch fixes this by implementing conversion of None to an element of L, which already works for most rings. There are other solutions, but those seem more complicated.

[pjbruin]

Commit: 64063cf

[pjbruin]

Author: Peter Bruin

[mwageringel]

comment:2

It seems comparison of ring morphisms with the identity morphism might be a wider problem. See #29632 for a similar issue.

[mwageringel]

comment:3

The None in the error comes from the generic implementation of _richcmp_ in sage.categories.morphism which performs a computation like this

sage: L.gen(0)._lmul_(K.gen(0)) is None
True

which looks like _lmul_ is not implemented. It seems that the implementation of the _richcmp_ method assumes that the result is

sage: L.gen(0) * K.gen(0)
b^3 + O(b^43)

so treating None as zero here will not correctly resolve this problem.

Another issue with the implementation of _richcmp_ is that the <ModuleElement>e, on which _lmul_ is called, might be zero if the last generator is 0. For example in this case:

sage: K.<a> = QuadraticField(-1)
sage: R.<x,y> = K['x,y'].quotient('y')
sage: id = R.Hom(R).identity()
sage: f = R.hom(R.gens(), R, base_map=K.hom([-a], K))
sage: f == id  # should be False
True

As e=y is zero, also y * a is zero, so that comparing the images f(y * a) and id(y * a) is not meaningful. (Currently, this example fails because _lmul_ returns None, though, and therefore also fails for the quotient by x.)

So at least the implementation should pick a module element e that is non-zero, but I am not sure if that is already a sufficient condition for checking equality.

[pjbruin]

comment:4

This _richcmp_() method was totally rewritten in #24281; adding the author and reviewer of that ticket in CC.

[pjbruin]

comment:5

Replying to @mwageringel:

It seems comparison of ring morphisms with the identity morphism might be a wider problem. See #29632 for a similar issue.

Possibly also related: #28617

[kwankyu]

comment:6

Replying to @mwageringel:

The None in the error comes from the generic implementation of _richcmp_ in sage.categories.morphism which performs a computation like this

sage: L.gen(0)._lmul_(K.gen(0)) is None
True

which looks like _lmul_ is not implemented. It seems that the implementation of the _richcmp_ method assumes that the result is

sage: L.gen(0) * K.gen(0)
b^3 + O(b^43)

so treating None as zero here will not correctly resolve this problem.

Right. Implementing scalar multiplication via _lmul_ for the ring elements will solve the problem.

[kwankyu]

comment:7

Now the patch for #28617 also fixes the problem of this ticket.

[pjbruin]

comment:8

Replying to @kwankyu:

Now the patch for #28617 also fixes the problem of this ticket.

With the patch from #28617 it no longer raises an error, but the answer is wrong:

sage: R.<x> = QQ[]
....: K = Qp(19)
....: L.<b> = K.extension(x^2 - 19)
....: a = L.hom([b], codomain=L)
....: id = L.Hom(L).identity()
....: a == id
False

The result should be True.

[kwankyu]

comment:9

Does a behave correctly?

sage: a(K.gen(0)) == K.gen(0)
False
sage: a(K.gen(0)) == L(K.gen(0))
False
sage: K.gen()
19 + O(19^21)
sage: L(K.gen(0))
b^2 + O(b^42)
sage: a(K.gen(0))
b^2 + 18*b^40 + O(b^42)

[pjbruin]

comment:10

This is funny. The output of L(K.gen(0)) is correct (since 19 = b^2 in L), but the output of a(K.gen(0)) has a wrong term 18*b^40... Same for a(b^2) (but not for a(b)):

sage: b
b + O(b^41)
sage: a(b)
b + O(b^41)
sage: b^2
b^2 + O(b^42)
sage: a(b^2)
b^2 + 18*b^40 + O(b^42)

[pjbruin]

comment:11

This looks suspicious (the term 18*19^20 should probably not be there):

sage: c = b^2
sage: c._polynomial_list()
[19 + 18*19^20 + O(19^21)]

And this:

sage: f, k = c._ntl_rep_abs(); f
[676619522235827247480400837]
sage: K(ZZ(f[0]))
19 + 18*19^20 + O(19^21)

[pjbruin]

comment:12

Relevant input for the above example:

sage: K = Qp(19)
sage: R.<x> = K[]
sage: L.<b> = K.extension(x^2 - 19)
sage: c = b^2
sage: f, k = c._ntl_rep_abs(); f
[676619522235827247480400837]
sage: K(ZZ(f[0]))
19 + 18*19^20 + O(19^21)

Maybe David Roe (CC) can say something about this?

[mkoeppe]

comment:13

Moving to 9.4, as 9.3 has been released.

[mkoeppe]

comment:14

Setting a new milestone for this ticket based on a cursory review.

[mkoeppe]

comment:15

Stalled in needs_review or needs_info; likely won't make it into Sage 9.5.