Polynomials over p-adics and power series are wrong

[videlec]

The design choice made for p-adics and power series is that O(p^n) and O(x^n) behave similarly to zero in comparisons. All of

• a.is_zero()
• not a
• a == a.parent().zero()
• not (a != a.parent().zero() answer True for O(p^n) or O(x^n)

sage: a = O(2^3)
sage: a.is_zero()
True
sage: a.is_zero() and not a and a == a.parent().zero() and not (a != a.parent().zero())
True
sage: R.<x> = PowerSeriesRing(QQ)
sage: a = O(x^3)
sage: a.is_zero() and not a and a == a.parent().zero() and not (a != a.parent().zero())
True

As a consequence, the generic implementation of uni-variate and multi-variate polynomials are broken on those rings (as it does remove terms whose coefficients are zero).

sage: Z2 = Zp(2)
sage: R.<x,y> = Z2[]
sage: R(1) - R(1)
0
sage: Z2(1) - Z2(1)
O(2^20)

Depends on #34000

Component: algebra

Issue created by migration from https://trac.sagemath.org/ticket/34024

[videlec]

comment:1

My personal feeling is that bool(a) should be equivalent to "a is not exact zero".