Singular

5.1.67 kbase

Syntax:

kbase ( ideal_expression )
kbase ( module_expression )
kbase ( ideal_expression, int_expression)
kbase ( module_expression, int_expression)

Type:

the same as the input type of the first argument

Purpose:

With one argument: computes a vector space basis (consisting of monomials) of the quotient ring by the ideal, resp. of a free module by the module, in case it is finite dimensional and if the input is a standard basis with respect to the ring ordering.
Note that, if the input is not a standard basis, the leading terms of the input are used and the result may have no meaning.
With two arguments: computes the part of a vector space basis of the respective quotient with degree of the monomials equal to the second argument. Here, the quotient does not need to be finite dimensional. If an attributeisHomog (of type intvec) is present, it is used as module weight.

Example:

ring r=32003,(x,y,z),ds; ideal i=x2,y2,z; kbase(std(i)); ==> _[1]=xy ==> _[2]=y ==> _[3]=x ==> _[4]=1 i=x2,y3,xyz; // quotient not finite dimensional kbase(std(i),2); ==> _[1]=z2 ==> _[2]=yz ==> _[3]=xz ==> _[4]=y2 ==> _[5]=xy