Ilya Zakharevich on Tue, 9 May 2000 23:51:11 -0400 |
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Re: Bug in Mod (2.0.15) |
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