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D.12.4.13 standard
Procedure from library teachstd.lib (see teachstd_lib).
- Usage:
- standard(i[,s]); id ideal or module, s int
- Return:
- a standard basis of id, using generalized Mora's algorithm
which is Buchberger's algorithm for global monomial orderings.
If s!=0 the symmetric s-polynomial (without division) is used
- Note:
- Show comments if printlevel > 0, pauses computation if printlevel > 1
Example:
| | LIB "teachstd.lib";
ring r=0,(x,y,z),dp;
ideal G = x2y+x2,y3+xyz,xyz2+z4;
standard(G);"";
==> _[1]=x2y+x2
==> _[2]=y3+xyz
==> _[3]=xyz2+z4
==> _[4]=x3z-x2y
==> _[5]=-xz4+x2z2
==> _[6]=-y2z4-x2z3
==> _[7]=z6+x2yz2
==> _[8]=x2z3-x3z
==> _[9]=-x2z2+x3
==> _[10]=x4-x2yz
==>
ring s=0,(x,y,z),(c,ds);
ideal G = 2x2+x2y,y3+xyz,3x3y+z4;
standard(G);"";
==> _[1]=2x2+x2y
==> _[2]=y3+xyz
==> _[3]=3x3y+z4
==> _[4]=-2/3z4+x3y2
==>
standard(G,1);""; //use symmetric s-poly without division
==> _[1]=2x2+x2y
==> _[2]=y3+xyz
==> _[3]=3x3y+z4
==> _[4]=-2z4+3x3y2
==>
module M = [2x2,x3y+z4],[3y3+xyz,y3],[5z4,z2];
standard(M);
==> _[1]=[2x2,x3y+z4]
==> _[2]=[3y3+xyz,y3]
==> _[3]=[5z4,z2]
==> _[4]=[0,-2/3x2y3+x3y4+1/3x4y2z+y3z4+1/3xyz5]
==> _[5]=[0,-2/5x2z2+x3yz4+z8]
==> _[6]=[0,-3/5y3z2-1/5xyz3+y3z4]
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