Basis of Quotient Ring

Sheng Mao · Thu Aug 11, 2005 5:32 pm

Let S be the polynomial ring of n- variables, J be a homogenous ideal generated by homogenous elements f_1,..,f_m. Consider the quotient ring S/J of S modulo J. Can one find a basis of the graded piece of (S/J)_i of given degree i by exploiting commands of SINGULAR? If so, what are the appropriate SINGULAR commands?

email: msheng@math.cuhk.edu.hk
Posted in old Singular Forum on: 2004-01-05 13:14:24+01

Re: Basis of Quotient Ring

levandov · Thu Aug 11, 2005 8:51 pm

Quote:

> Let S be the polynomial ring of n- variables, J be a homogenous ideal generated by homogenous elements f_1,..,f_m.

>Consider the quotient ring S/J of S modulo J. Can one find a basis of the graded piece of (S/J)_i of given degree i by exploiting commands of SINGULAR? If so, what are the appropriate SINGULAR commands?

There are two commands you can use: kbase and jet.
We recommend to use kbase with the second argument, for example

Code:

ring r=32003,(x,y,z),ds;
ideal i=x2,y3,xyz;
kbase(std(i),2);
==> _[1]=z2
==> _[2]=yz
==> _[3]=xz
==> _[4]=y2
==> _[5]=xy

More information you can find in the documentation.

kbase: http://www.singular.uni-kl.de/Manual/3-0-0/sing_218.htm

jet: http://www.singular.uni-kl.de/Manual/3-0-0/sing_217.htm

Have fun,