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D.8.4.3 triangM
Procedure from library triang.lib (see triang_lib).
- Usage:
- triangM(G[,i]); G=ideal, i=integer,
- 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).
If i = 2, then each polynomial of the triangular systems
is factorized.
- Note:
- Algorithm of Moeller (see: Moeller, H.M.: On decomposing systems of
polynomial equations with finitely many solutions, Appl. Algebra Eng.
Commun. Comput. 4, 217 - 230, 1993).
Example:
| | LIB "triang.lib";
ring rC5 = 0,(e,d,c,b,a),lp;
triangM(stdfglm(cyclic(5))); //oder: triangM(stdfglm(cyclic(5)),2);
==> [1]:
==> _[1]=a5-1
==> _[2]=b-a
==> _[3]=c-a
==> _[4]=d2+3da+a2
==> _[5]=e+d+3a
==> [2]:
==> _[1]=a10+123a5+1
==> _[2]=55b2-2ba6-233ba-8a7-987a2
==> _[3]=55c+a6+144a
==> _[4]=55d+a6+144a
==> _[5]=55e+55b-2a6-233a
==> [3]:
==> _[1]=a5-1
==> _[2]=b6+4b5a+5b4a2+5b3a3+5b2a4+4b+a
==> _[3]=5c+8b5a+30b4a2+30b3a3+25b2a4+30b+22a
==> _[4]=5d-2b5a-10b4a2-15b3a3-10b2a4-10b-8a
==> _[5]=5e-6b5a-20b4a2-15b3a3-15b2a4-15b-9a
==> [4]:
==> _[1]=a5-1
==> _[2]=b-a
==> _[3]=c2+3ca+a2
==> _[4]=d+c+3a
==> _[5]=e-a
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