Remembering basis transformations

Jerry · Thu Oct 27, 2011 5:55 pm

If I take some ideal I=(f_1,...,f_n) and then generate it by a Groebner basis I=(g_1,...,g_m) is there some way to automatically have it also return how to express g_i's in terms of f_i's?

This information is at least implicitly contained in the computation of the Groebner basis but I was wonder if there is any explicit way to extract it in plural.

Thanks.

Re: Remembering basis transformations

gorzel · Fri Oct 28, 2011 12:39 pm

Use the command lift
http://www.singular.uni-kl.de/Manual/3-0-4/sing_235.htm

Code:

> ring r = 0,(x,y,z),dp;
> ideal I=3x2+yz,7y6+2x2y,4x2y3-5xz;
> ideal J = std(I);
> J;
J[1]=3x2+yz
J[2]=8xz2-105yz2
J[3]=33075y2z2+64yz3
J[4]=315xy2z+8y2z2
J[5]=4y4z+15xz
J[6]=8388608yz5-56987567117578125yz2
J[7]=21y6-2y2z
> matrix T = lift(J,I);
> matrix(J)*T;       // This is I again, now as a matrix
_[1,1]=3x2+yz
_[1,2]=7y6+2x2y
_[1,3]=4x2y3-5xz
> _ == I;
1
> I;
I[1]=3x2+yz
I[2]=7y6+2x2y
I[3]=4x2y3-5xz

Note: lift makes again a std-basis computation; there is no direct
way to obtain automatically the transformation matrix
from the initial std computation.

Re: Remembering basis transformations

hannes · Fri Oct 28, 2011 5:42 pm

liftstd produces both: the standard basis and the transformation matrix

Re: Remembering basis transformations

Jerry · Tue Nov 08, 2011 7:14 pm

Thanks for the responses.

Is there also a way to do it with twostd()?