Equality in PowerSeriesRing can be unpredictable

[darij]

I know this is unavoidable, but I'd say it should be better documented...

Over a finite field:

sage: K.<u> = PowerSeriesRing(GF(5))
sage: u = K.gen()
sage: u ** 25
u^25
sage: u ** 25 == 0
False
sage: (1 + u) ** 25
1 + u^25
sage: (1 + u) ** 25 == 1
False
sage: (1 / (1 - u)) ** 25     
1 + O(u^20)
sage: (1 / (1 - u)) ** 25 == 1
True

I suspect the last True is due to 1 / (1 - u) being a dense series, which leads to Sage keeping precision at O(u^20) rather than moving to a higher exponent.

[vbraun]

The power series ring tries to keep arbitrary precision if possible, and only switches to series expansion if necessary. So u^25 is kept as exact, even though it is zero to the default_prec=20.

For interactive use, I think this is what one would want. However, for non-interactive use an "always approximate" mode where u^25 is immediately zero would be nice. I have written lots of code where many lines end in +O(u^prec) to get the precision back down.