Hi Bill,

> Fredrik and I found the following in Sage 6.3.
> sage: R.<x> = PowerSeriesRing(ZZ)
> sage: S.<y> = PowerSeriesRing(R)
> sage: x + O(x^4) + y + O(y^4)
> x + O(x^4) + y + O(y^4)
> sage: y + O(y^4) + x + O(x^4)
> x + y + O(y^4)
> I guess this comes from Pari defining:
> 0 == O(x^4)
> to be true.

Sage currently implements power series on top of polynomials, not via
PARI (but I have been working on a different implementation based on
PARI series, see http://trac.sagemath.org/ticket/15601).

Peter

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