|
5.1.129 stdhilb
Procedure from library standard.lib (see standard_lib).
- Syntax:
stdhilb ( ideal_expression )
stdhilb ( module_expression )
stdhilb ( ideal_expression, intvec_expression )
stdhilb ( ideal_expression, list of string_expressions
and intvec_expressin )
- Type:
- type of the first argument
- Purpose:
- Compute a Groebner basis of the ideal/module in the basering by
using the Hilbert driven Groebner basis algorithm.
If an argument of type string
"std" resp. "slimgb"
is given, the standard basis computation uses std or
slimgb , otherwise a heuristically chosen method (default)
- Theory:
- If the ideal is not homogeneous compute first a Groebner basis
of the homogenization of the ideal, then the Hilbert function and,
finally, a Groebner basis in the original ring by using the
computed Hilbert function.
If the ideal is homogeneous and a second argument of type intvec
is given it will be used as 1st Hilbert function in the Hilbert
driven algorithm.
- Note:
- 'homogeneous' means weighted homogeneous with respect to the weights
w[i] of the variables var(i) of the basering. Parameters are not
converted to variables.
- Assume:
- The argument of type intvec is the 1st Hilbert series, computed
by
hilb using an intvector w, w[i]=deg(var(i)), as third
argument
Example:
| ring r = 0,(x,y,z),lp;
ideal i = y3+x2,x2y+x2z2,x3-z9,z4-y2-xz;
ideal j = stdhilb(i); j;
==> j[1]=z10
==> j[2]=yz9
==> j[3]=2y2z4-z8
==> j[4]=2y3z3-yz7-z9
==> j[5]=y4+y3z2
==> j[6]=xz+y2-z4
==> j[7]=xy2-xz4-y3z
==> j[8]=x2+y3
ring r1 = 0,(x,y,z),wp(3,2,1);
ideal i = y3+x2,x2y+x2z2,x3-z9,z4-y2-xz; //ideal is homogeneous
ideal j = stdhilb(i,"std"); j;
==> j[1]=y2+xz-z4
==> j[2]=x2+y3
==> j[3]=2xz5-z8
==> j[4]=2xyz4+y2z5-xz6-yz7+z9
==> j[5]=z10
==> j[6]=2yz9+z11
//this is equivalent to:
intvec v = hilb(std(i),1);
ideal j1 = std(i,v,intvec(3,2,1)); j1;
==> j1[1]=y2+xz-z4
==> j1[2]=x2-xyz+yz4
==> j1[3]=2xz5-z8
==> j1[4]=2xyz4-yz7+z9
==> j1[5]=z10
==> j1[6]=yz9
size(NF(j,j1))+size(NF(j1,j)); //j and j1 define the same ideal
==> 0
| stdfglm, std, slimgb, groebner
|