Hi Simon, please do this by hand. for I=R*(4*x^2*y^2+2*x*y^3+3*x*y, 2*x^2+x*y, 2*y^2) we get f=y*I.0 -2*y^3*I.1 -x*I.2 = x*y^2 We CAN reduce f on [x*y^3, 2*x^2 + x*y, 3*x*y, 2*y^2] to zero because f = y * (3*x*y) - x * (2*y^2) . So this gives that [x*y^3, 2*x^2 + x*y, 3*x*y, 2*y^2] is Groebner basis (which minimal form is [2*x^2 + x*y, 3*x*y, 2*y^2]. So toy and libsingular:slimgb are correct algorithms. What do you think?
On Fri, Jul 30, 2010 at 2:51 PM, Simon King <[email protected]> wrote: > Hi Dusan, > > On Jul 30, 12:58 pm, Dušan Orlović <[email protected]> wrote: > > Hi Simon, thanks for replying, > > your example: y*I.0 -2*y^3*I.1 -x*I.2 is x*y^2 which lies in ideal > [x*y^3, > > 2*x^2 + x*y, 3*x*y, > > 2*y^2] as y * (3*x*y) - x * (2*y^2) = x*y^2, so it has to be reduced to > > zero. Am I right? > > Yes. I wrote "y*I.0 -2*y^3*I.1 -x*I.2" in order to demonstrate that we > really have an element of the ideal that is not reduced to zero, in > order to assert that the answer you expected was not a Groebner basis > in degree reverse lexicographic order (and x>y, by the way). > > > Please tell me which command should I use to avoid bugs. > > Is it the parameter (algorithm='toy:buchberger2') ? > > As the name says, 'toy:...' is a toy, hardly useful for any serious > computation. And, as I demonstrated in my previous post, the answer > given with toy:buchberger and toy:buchberger2 and (surprisingly) > libsingular:slimgb are wrong. > > It is hard to say how to avoid bugs. Groebner bases over the integers > are relatively new in Singular (hence, also new in Sage), and > therefore it does not surprise me that there are still a couple of > bugs to discover. > > But I don't know where the bug is actually located. Theoretically, it > could also be in "reduce". > > The best way to avoid bugs? Well, one should never blindly trust any > computer algebra system. Doing a verification by hand (as you did) is > certainly a good approach, as long as the examples are small enough. > > I just arrived in my office, I am now doing some tests, also involving > other computer algebra systems. I'll report back when I opened a trac > ticket for this. > > Cheers, > Simon > > -- > To post to this group, send an email to [email protected] > To unsubscribe from this group, send an email to > [email protected]<sage-devel%[email protected]> > For more options, visit this group at > http://groups.google.com/group/sage-devel > URL: http://www.sagemath.org > -- To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
