Refactor O() and fix O() for lazy power series ring

[user202729]

Fixes O(x) for LazyPowerSeriesRing. (previously it doesn't work)

Also refactor it to check less special cases.

Use .perfect_power() instead of factor() to avoid wasting time if a large composite is passed.

Note: .O() and .add_bigoh() is the same for many rings, but not AsymptoticRing.
In particular

• for most rings x.O(5) means x.add_bigoh(5) or x+O(x^5)
• for AsymptoticRing x.O() means O(x)

Personally I think the latter makes more sense, but this is part of public API so we will have to live with this inconsistency. (The former does not make sense for AsymptoticRing even)

sage: A.<n> = AsymptoticRing(growth_group='QQ^n * n^QQ * log(n)^QQ', coefficient_ring=QQ); A
Asymptotic Ring <QQ^n * n^QQ * log(n)^QQ * Signs^n> over Rational Field
sage: n.O()
O(n)
sage: R.<x> = QQ[[]]
sage: x.O()
Traceback (most recent call last)
...
TypeError: O() takes exactly 1 positional argument (0 given)
sage: x.O(3)
x + O(x^3)

Thus the explicit check is needed. (add_bigoh is not implemented for AsymptoticRing)

I don't know why it's desirable to block O(2*x^2) for polynomial ring, but I'll leave the behavior unchanged just in case.

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[user202729]

Looks like genuine bug in doctest.

sage: R.<x> = ZZ[]
sage: W.<w> = ZpFM(5).extension(x^3 - 5)
sage: (1 + w)._polynomial_list()
[1, 1]
sage: (1 + w + O(w^11))._polynomial_list(pad=True)
[1, 1, 0]

The problem is the extension is ramified (Eisenstein), so O(…) is not yet implemented (sort of). (i.e. O(w^11) is identical to 0). It would actually be misleading to write the test like that.

But then, add_bigoh isn't implemented for fixed modulus in general either (even if the extension is unramified). So writing +O(w^…) is misleading in general in that it does nothing.

sage: W.<w> = ZpFM(5).extension(x^3 - 5)
sage: w^5
w^5
sage: O(w^4)  # before this change, gives error after this change
0
sage: w^5 + O(w^4)
w^5
sage: 1 + w + O(w^11)
1 + w

[tscrim]

LGTM except I don't know the math enough to verify the padic change. @xcaruso Can you comment on that doctest?

[tscrim]

@xcaruso @roed314 Could one of you comment on the changed doctest and/or analysis? I don't have the expertise to verify.

[user202729]

A (more conservative) alternative is to implement a check in implementation of O to return zero for fixed modulus rings (so, another special case apart from Puiseux series).

But I really can't think of any case where O(something) in fixed modulus ring makes sense, since writing +O(…) does nothing, and even in generic code it seems not particularly useful. Might be better to raise an error to warn the user.

[roed314]

I'll take a look, though probably won't have time for a couple days.