Singular Manual: kbase

  | ### 5.1.69 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 attribute`isHomog` (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
See ideal; module; vdim.

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