Singular

D.8.4.4 triangMH

Usage:

triangMH(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 disjoint 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 and Hillebrand (see: Moeller, H.M.: On decomposing systems of polynomial equations with finitely many solutions, Appl. Algebra Eng. Commun. Comput. 4, 217 - 230, 1993 and Hillebrand, D.: Triangulierung nulldimensionaler Ideale - Implementierung und Vergleich zweier Algorithmen, master thesis, Universitaet Dortmund, Fachbereich Mathematik, Prof. Dr. H.M. Moeller, 1999).

LIB "triang.lib";
ring rC5 = 0,(e,d,c,b,a),lp;
triangMH(stdfglm(cyclic(5)));
==> [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