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5.1.124 stdhilb

Procedure from library standard.lib (see standard_lib).

Syntax:
stdhilb ( ideal_expression )
stdhilb ( ideal_expression, intvec_expression )

Type:
ideal

Purpose:
Computes the standard basis of the given ideal in the basering, via a Hilbert driven standard basis computation.
If the ideal is homogeneous then the optional second argument will be used as 1st Hilbert function.
If the ideal is not homogeneous and if the ordering is either dp or non-global then std is called. For other orderings, stdhilb computes the Hilbert series of the homogenized ideal (by applying std in a ring with dp ordering) and uses the result to navigate the standard basis computation with respect to the given ordering.

Assume:
The optional second argument is the output of hilb(j,1), where j is a standard basis for the given (homogeneous) ideal.

Example:
 
   ring  r=0,(w,x,y,z),lp;
ideal i=y3+x2w,x2y+x2w,x3-x2w,z4-x2w2-yw3;
ideal i1=stdhilb(i); i1;
==> i1[1]=z12
==> i1[2]=yz8
==> i1[3]=y2z4
==> i1[4]=xy3+y4
==> i1[5]=x2z4
==> i1[6]=x2y-y3
==> i1[7]=x3+x2y
==> i1[8]=wy3-x3y
==> i1[9]=wx2-x3
==> i1[10]=w3y+w2x2-z4
// the latter computation is equivalent to:
ring  R=0,(w,x,y,z),dp;
ideal i=y3+x2w,x2y+x2w,x3-x2w,z4-x2w2-yw3;
intvec v=hilb(std(i),1);
setring r;      
ideal i2=std(i,v); i2;
==> i2[1]=z12
==> i2[2]=yz8
==> i2[3]=y2z4
==> i2[4]=xy3+y4
==> i2[5]=x2z4
==> i2[6]=x2y-y3
==> i2[7]=x3+x2y
==> i2[8]=wy3-x3y
==> i2[9]=wx2-x3
==> i2[10]=w3y+w2x2-z4
See also: groebner; std; stdfglm.


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