Re: Bug in Mod (2.0.15)

[Ilya Zakharevich <ilya@math.ohio-state.edu>]

On Fri, May 05, 2000 at 01:51:56PM -0400, I wrote:
> > >Given marked generators (x,y,z etc) of the ring, could not we get
> > >something canonical?
> > 
> > I check it in a book:
> > No, but does it really matter ?
> 
> Yes and Yes.  Why it is possible:

> [...] This gives a canonical basis (at least for polynomials over a field,
> but probably over some rings too), and I thought it is the same
> procedure as for Groebner bases.

It looks like what I wrote is exactly the "minimal" Groebner basis
(by inclusion) [if what I understood is correct].  The leading
monomials are those which are not proportional to other leading
monomials from the ideal.

> > We only need to be able to: 
> > 
> > _ check wether an element is in an ideal or not.
> > _ Reduce an element modulo an ideal to avoid coefficient explosion.
> > _ Compute inverse if it exists.
> > 
> > and Groebner basis can handle this.
> 
> We also need to able to compare things for equality.

But you are right: it is possible to do operations with any

  Mod(Mod(Mod(A, P1...),Pn))

But the assumption that Pk are canonical may speed up some operations.

Ilya