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D.8.4.2 triangLfak
Procedure from library triang.lib (see triang_lib).
- Usage:
- triangLfak(G); G=ideal
- Assume:
- G is the reduced lexicographical Groebner basis of the
zero-dimensional ideal (G), sorted by increasing leading terms.
- Return:
- a list of finitely many triangular systems, such that
the union of their varieties equals the variety of (G).
- Note:
- Algorithm of Lazard with factorization (see: Lazard, D.: Solving
zero-dimensional algebraic systems, J. Symb. Comp. 13, 117 - 132, 1992).
- Remark:
- each polynomial of the triangular systems is factorized.
Example:
| | LIB "triang.lib";
ring rC5 = 0,(e,d,c,b,a),lp;
triangLfak(stdfglm(cyclic(5)));
==> [1]:
==> _[1]=a-1
==> _[2]=b-1
==> _[3]=c-1
==> _[4]=d2+3d+1
==> _[5]=e+d+3
==> [2]:
==> _[1]=a-1
==> _[2]=b-1
==> _[3]=c2+3c+1
==> _[4]=d+c+3
==> _[5]=e-1
==> [3]:
==> _[1]=a-1
==> _[2]=b4+b3+b2+b+1
==> _[3]=-c+b2
==> _[4]=-d+b3
==> _[5]=e+b3+b2+b+1
==> [4]:
==> _[1]=a-1
==> _[2]=b2+3b+1
==> _[3]=c+b+3
==> _[4]=d-1
==> _[5]=e-1
==> [5]:
==> _[1]=a4+a3+a2+a+1
==> _[2]=b-1
==> _[3]=c+a3+a2+a+1
==> _[4]=-d+a3
==> _[5]=-e+a2
==> [6]:
==> _[1]=a4+a3+a2+a+1
==> _[2]=b-a
==> _[3]=c-a
==> _[4]=d2+3da+a2
==> _[5]=e+d+3a
==> [7]:
==> _[1]=a4+a3+a2+a+1
==> _[2]=b-a
==> _[3]=c2+3ca+a2
==> _[4]=d+c+3a
==> _[5]=e-a
==> [8]:
==> _[1]=a4+a3+a2+a+1
==> _[2]=b3+b2a+b2+ba2+ba+b+a3+a2+a+1
==> _[3]=c+b2a3+b2a2+b2a+b2
==> _[4]=-d+b2a2+b2a+b2+ba2+ba+a2
==> _[5]=-e+b2a3-ba2-ba-b-a2-a
==> [9]:
==> _[1]=a4+a3+a2+a+1
==> _[2]=b2+3ba+a2
==> _[3]=c+b+3a
==> _[4]=d-a
==> _[5]=e-a
==> [10]:
==> _[1]=a4-4a3+6a2+a+1
==> _[2]=-11b2+6ba3-26ba2+41ba-4b-8a3+31a2-40a-24
==> _[3]=11c+3a3-13a2+26a-2
==> _[4]=11d+3a3-13a2+26a-2
==> _[5]=-11e-11b+6a3-26a2+41a-4
==> [11]:
==> _[1]=a4+a3+6a2-4a+1
==> _[2]=-11b2+6ba3+10ba2+39ba+2b+16a3+23a2+104a-24
==> _[3]=11c+3a3+5a2+25a+1
==> _[4]=11d+3a3+5a2+25a+1
==> _[5]=-11e-11b+6a3+10a2+39a+2
==> [12]:
==> _[1]=a2+3a+1
==> _[2]=b-1
==> _[3]=c-1
==> _[4]=d-1
==> _[5]=e+a+3
==> [13]:
==> _[1]=a2+3a+1
==> _[2]=b+a+3
==> _[3]=c-1
==> _[4]=d-1
==> _[5]=e-1
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