SINGULAR                             /
 A Computer Algebra System for Polynomial Computations   /   version 2-0-1
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     by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   June 2001
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
// ** loaded /local/math/Singular/2-0-1/LIB/primdec.lib (1.98.2.4,2001/06/18)
// ** loaded /local/math/Singular/2-0-1/LIB/matrix.lib (1.26,2001/01/18)
// ** loaded /local/math/Singular/2-0-1/LIB/ring.lib (1.17,2001/01/16)
// ** loaded /local/math/Singular/2-0-1/LIB/inout.lib (1.21,2001/01/16)
// ** loaded /local/math/Singular/2-0-1/LIB/random.lib (1.16,2001/01/16)
// ** loaded /local/math/Singular/2-0-1/LIB/poly.lib (1.33,2001/01/16)
// ** loaded /local/math/Singular/2-0-1/LIB/elim.lib (1.14,2001/01/16)
// ** loaded /local/math/Singular/2-0-1/LIB/general.lib (1.38.2.1,2001/04/24)
This is the degree of the original ideal; which has irrelevant components
as well as relevant ones that we wish to study
// codimension = 6
// dimension   = 4
// degree      = 8
We saturate by the ideal of rank 1 quadrics
// codimension = 7
// dimension   = 3
// degree      = 20
Now we saturate by an ideal defining a trivial 2,2 curve
Saturate by (g,f,e,d,c,b,a).
// codimension = 7
// dimension   = 3
// degree      = 10
Now we saturate by an ideal defining a trivial 2,2 curve
Saturate by (y,x,h,g,f,d,c).
// codimension = 8
// dimension   = 2
// degree      = 24
Thus we have finaly obtain an ideal of dimension 2, so that it
defines curves.
The first component of the ideal has degree 4, and is reducible, but
does not contain any real points
_[1]=3*x^2+3*x*y+y^2
_[2]=h+2*x+y
_[3]=f+g
_[4]=e*y^2-3*g^2*x-3*g^2*y
_[5]=e*x+g^2
_[6]=d*y-6*g*x-3*g*y
_[7]=d*x+3*g*x+2*g*y
_[8]=d*g-2*e*y+3*g^2
_[9]=d^2+3*g^2
_[10]=c+g
_[11]=b*y-2*e*y+3*g^2
_[12]=b*x+e*y-g^2
_[13]=2*b*g-d*e-e*g
_[14]=2*b*d-d*e+3*e*g
_[15]=b^2-b*e+e^2
_[16]=a-2*b+e
// codimension = 8
// dimension   = 2
// degree      = 4
/////////////////////////////////////////
The same is true for the second component
_[1]=3*x^2+3*x*y+y^2
_[2]=h+2*x+y
_[3]=f+g
_[4]=e*y^2-3*g^2*x-3*g^2*y
_[5]=e*x+g^2
_[6]=d*y+6*g*x+3*g*y
_[7]=d*x-3*g*x-2*g*y
_[8]=d*g+2*e*y-3*g^2
_[9]=d^2+3*g^2
_[10]=c-g
_[11]=b*y+2*e*y-3*g^2
_[12]=b*x-e*y+g^2
_[13]=2*b*g-d*e+e*g
_[14]=2*b*d+d*e+3*e*g
_[15]=b^2+b*e+e^2
_[16]=a+2*b+e
// codimension = 8
// dimension   = 2
// degree      = 4
/////////////////////////////////////////
And the third component
_[1]=3*x^2-3*x*y+y^2
_[2]=h-2*x+y
_[3]=f-g
_[4]=e*y^2+3*g^2*x-3*g^2*y
_[5]=e*x-g^2
_[6]=d*y-6*g*x+3*g*y
_[7]=d*x-3*g*x+2*g*y
_[8]=d*g+2*e*y-3*g^2
_[9]=d^2+3*g^2
_[10]=c+g
_[11]=b*y+2*e*y-3*g^2
_[12]=b*x+e*y-g^2
_[13]=2*b*g-d*e+e*g
_[14]=2*b*d+d*e+3*e*g
_[15]=b^2+b*e+e^2
_[16]=a+2*b+e
// codimension = 8
// dimension   = 2
// degree      = 4
/////////////////////////////////////////
And the fourth component
_[1]=3*x^2-3*x*y+y^2
_[2]=h-2*x+y
_[3]=f-g
_[4]=e*y^2+3*g^2*x-3*g^2*y
_[5]=e*x-g^2
_[6]=d*y+6*g*x-3*g*y
_[7]=d*x+3*g*x-2*g*y
_[8]=d*g-2*e*y+3*g^2
_[9]=d^2+3*g^2
_[10]=c-g
_[11]=b*y-2*e*y+3*g^2
_[12]=b*x-e*y+g^2
_[13]=2*b*g-d*e-e*g
_[14]=2*b*d-d*e+3*e*g
_[15]=b^2-b*e+e^2
_[16]=a-2*b+e
// codimension = 8
// dimension   = 2
// degree      = 4
/////////////////////////////////////////
The last 4 components each have degree 2 and are defined over the
real numbers, defining RATIONAL components !
_[1]=x
_[2]=h+y
_[3]=g
_[4]=f
_[5]=d
_[6]=c^2+4*e*y
_[7]=b
_[8]=a-3*e
// codimension = 8
// dimension   = 2
// degree      = 2
/////////////////////////////////////////
_[1]=x
_[2]=h+y
_[3]=g
_[4]=f
_[5]=d^2+4*e*y
_[6]=c
_[7]=b
_[8]=a+e
// codimension = 8
// dimension   = 2
// degree      = 2
/////////////////////////////////////////
_[1]=x
_[2]=3*h-y
_[3]=g
_[4]=4*e*y-3*f^2
_[5]=d
_[6]=c
_[7]=b
_[8]=a-3*e
// codimension = 8
// dimension   = 2
// degree      = 2
/////////////////////////////////////////
_[1]=x
_[2]=3*h-y
_[3]=f
_[4]=4*e*y-3*g^2
_[5]=d
_[6]=c
_[7]=b
_[8]=a+e
// codimension = 8
// dimension   = 2
// degree      = 2
/////////////////////////////////////////
10
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