Thanks everyone for the discussion.
I opened https://trac.sagemath.org/ticket/24071 to improve the
docstring of reduce().
It's ready for review, let's continue the discussion there.
> In Singular's implementation p' does not depend on the order of the
> polynomials in I because it starts with sorting I
> (wrt. to the monomial ordering or the total degree)."
I had come to the same conclusion, but that's not a total ordering, so:
sage: A.<x,y> = QQ[]
sage: (x+y).reduce([x+y, x-y])
2*y
sage: (x+y).reduce([x-y, x+y])
0
Luca
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