Singular

D.2.4.10 ConsLevels

Usage:

ConsLevels(list L);
L=[[P1,Q1],...,[Ps,Qs]] is a list of lists of of pairs of ideals represening the constructible set
S=V(P1) \ V(Q1) u ... u V(Ps) \ V(Qs).
To be called in a ring Q[a][x] or a ring Q[a]. But the ideals can contain only the parameters in Q[a].

Return:

The list of ideals [a1,a2,...,at] representing the
closures of the canonical levels of S and its
complement C wrt to the closure of S. The
canonical levels of S are represented by theirs
Crep. So we have:
Levels of S: [a1,a2],[a3,a4],...
Levels of C: [a2,a3],[a4,a5],...
S=V(a1) \ V(a2) u V(a3) \ V(a4) u ...
C=V(a2 \ V(a3) u V(a4) \ V(a5) u ...
The expression of S can be obtained from the
output of ConsLevels by
the call to Levels.

Note:

The algorithm was described in
J.M. Brunat, A. Montes. "Computing the canonical
representation of constructible sets."
Math. Comput. Sci. (2016) 19: 165-178.

LIB "grobcov.lib";
if(defined(R)){kill R;}
ring R=0,(x,y,z),lp;
short=0;
ideal P1=x*(x^2+y^2+z^2-1);
ideal Q1=z,x^2+y^2-1;
ideal P2=y,x^2+z^2-1;
ideal Q2=z*(z+1),y,x*(x+1);
list Cr1=Crep(P1,Q1);
list Cr2=Crep(P2,Q2);
list L=list(Cr1,Cr2);
L;
==> [1]:
==>    [1]:
==>       _[1]=x^3+x*y^2+x*z^2-x
==>    [2]:
==>       _[1]=z
==>       _[2]=x^2+y^2-1
==> [2]:
==>    [1]:
==>       _[1]=y
==>       _[2]=x^2+z^2-1
==>    [2]:
==>       _[1]=z^2+z
==>       _[2]=y
==>       _[3]=x+z+1
ConsLevels(L);
==> [1]:
==>    _[1]=x^3+x*y^2+x*z^2-x
==> [2]:
==>    _[1]=z
==>    _[2]=x^2+y^2-1
==> [3]:
==>    _[1]=z
==>    _[2]=y
==>    _[3]=x-1
==> [4]:
==>    _[1]=1