Singular

D.2.4.11 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 ...
It is used internally by locus procedure.
The expression of S can be obtained from the output of ConsLevels by the call to ConsLevelsToLevels.

Note:

Th ring can be Q[a][x] or Q[a], but the ideals can only contain parmeters in Q[a]. The algorithm is 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