Singular

D.8.4.1 triangL

Procedure from library `triang.lib` (see triang_lib).

Usage:
triangL(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 (see: Lazard, D.: Solving zero-dimensional algebraic systems, J. Symb. Comp. 13, 117 - 132, 1992).

Example:
 ```LIB "triang.lib"; ring rC5 = 0,(e,d,c,b,a),lp; triangL(stdfglm(cyclic(5))); ==> [1]: ==> _[1]=a5-1 ==> _[2]=b-a ==> _[3]=c2+3ca+a2 ==> _[4]=d+c+3a ==> _[5]=e-a ==> [2]: ==> _[1]=a5-1 ==> _[2]=b-a ==> _[3]=c-a ==> _[4]=d2+3da+a2 ==> _[5]=e+d+3a ==> [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]=a10+123a5+1 ==> _[2]=55b2-2ba6-233ba-8a7-987a2 ==> _[3]=55c+a6+144a ==> _[4]=55d+a6+144a ==> _[5]=55e+55b-2a6-233a ```