Cached methods with do_pickle=True for UniqueRepresentation

[kwankyu]

fixes #28096.

The doctest failure https://github.com/sagemath/sage/actions/runs/17940271644/job/51014975901?pr=40874 is not related with this PR branch.

What the PR branch does is explained clearly in #28096 (comment)

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[kwankyu]

About this comment: #28096 (comment) I think that there is no side effect except negligible increase of pickle size (due to added empty dict) for unique representation objects.

[vincentmacri]

Thanks for getting this ready! This fixes a very subtle bug so I want to test locally (which will take a bit longer). I'll try to get to this in the next week or so.

I'm guessing from the genus function in your doctest that we might have encountered this bug in the same way (I found it while trying to cache the genus of a function field).

[vincentmacri]

About this comment: #28096 (comment) I think that there is no side effect except negligible increase of pickle size (due to added empty dict) for unique representation objects.

I'm going to test this anyway, but I assume this doesn't change the ability to load in pickles from before this change?

[kwankyu]

... I assume this doesn't change the ability to load in pickles from before this change?

I don't know. We need to test.

[kwankyu]

... I'll try to get to this in the next week or so.

From the next week or so, I am out of town for a week or two. So take your time.

[kwankyu]

... I assume this doesn't change the ability to load in pickles from before this change?

I don't know. We need to test.

I tested. It is backward compatible.

On the develop branch:

sage: P.<a,b,c> = PolynomialRing(QQ,3, order='lex')
sage: I = sage.rings.ideal.Katsura(P,3)  
sage: I.groebner_basis()                     # @cached_method(do_pickle=True)
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1,
 b + 30*c^3 - 79/7*c^2 + 3/7*c,
 c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
sage: I.save('test1')
sage: UCF = UniversalCyclotomicField()       # UniqueRepresentation
sage: UCF.zero()                             # @cached_method
0
sage: UCF.save('test2')                                                                                                                                  

Then on the PR branch:

sage: Ip = load('test1')
sage: Ip.groebner_basis.cache
{(('', None, None, False),
  ()): [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1,
  b + 30*c^3 - 79/7*c^2 + 3/7*c,
  c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]}
sage: P.<a,b,c> = PolynomialRing(QQ,3, order='lex')
sage: I = sage.rings.ideal.Katsura(P,3)
sage: I.groebner_basis.cache
{}
sage: Ip is I
False
sage: Ip == I
True
sage: UCFp  = load('test2')
sage: UCF = UniversalCyclotomicField()
sage: UCFp is UCF
True
sage: UCFp.zero.cache
sage: UCFp.zero()
0
sage: UCFp.zero.cache
0

By the way, I could not find a class with UniqueRepresentation and @cached_method(do_pickle=True). Of course, because it did not work.