On Tuesday, October 17, 2017 at 4:50:20 AM UTC-5, Luca De Feo wrote:
>
> > It takes I as the generators of the ideal and uses that as the reduction
> > set.
>
> That's not a definition. I'm in front of a class asking what this
> function does, and I'm unable to give a mathematical definition of
> what Sage means by "reduction" modulo something that's not a Groebner
> basis.
>
> My understanding is it does the (naïve?) reduction algorithm:
reducing = True
while reducing:
reducing = False
for expr in I:
if expr.leading_monomial().divides(cur.leading_monomial()):
cur -= cur.leading_term() / expr.leading_term() * expr
reducing = True
So this is well-defined, but it doesn't guarantee a unique result: it
depends on the ordering of I.
I agree with Martin, that this could have unintended consequences with
doing things in quotient rings because IIRC this is the key function that
is called for every operation. So having something that checks that the
input is a GB could slow things down. IMO, it also makes sense to expose
this to users.
Best,
Travis
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