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D.2.4.2 cgsdr

Procedure from library grobcov.lib (see grobcov_lib).

Return:
Returns a list T describing a reduced and disjoint comprehensive Groebner system (CGS), and whose segments correspond to constant leading power products (lpp) of the reduced Groebner basis. The returned list is of the form: ( (lpp, (basis,segment),...,(basis,segment)), ..,, (lpp, (basis,segment),...,(basis,segment)) ) The bases are the reduced Groebner bases (after normalization) for each point of the corresponding segment.
Each segment is given by a reduced representation (Ni,Wi), with Ni radical and V(Ni)=Zariski closure of the segment Si=V(Ni)\V(hi), where hi is the product of the polynomials w in Wi.

Note:
The basering R, must be of the form Q[a][x], a=parameters, x=variables, and should be defined previously, and the ideal defined on R.

Example:
 
LIB "grobcov.lib";
"Casas conjecture for degree 4";
==> Casas conjecture for degree 4
ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp;
ideal F=x1^4+(4*a3)*x1^3+(6*a2)*x1^2+(4*a1)*x1+(a0),
x1^3+(3*a3)*x1^2+(3*a2)*x1+(a1),
x2^4+(4*a3)*x2^3+(6*a2)*x2^2+(4*a1)*x2+(a0),
x2^2+(2*a3)*x2+(a2),
x3^4+(4*a3)*x3^3+(6*a2)*x3^2+(4*a1)*x3+(a0),
x3+(a3);
cgsdr(F);
==> [1]:
==>    [1]:
==>       _[1]=1
==>    [2]:
==>       [1]:
==>          [1]:
==>             _[1]=1
==>          [2]:
==>             _[1]=0
==>          [3]:
==>             _[1]=(a0-4*a1*a3+6*a2*a3^2-3*a3^4)
==>       [2]:
==>          [1]:
==>             _[1]=1
==>          [2]:
==>             _[1]=(a0-4*a1*a3+6*a2*a3^2-3*a3^4)
==>          [3]:
==>             _[1]=(a1-3*a2*a3+2*a3^3)
==>             _[2]=(16*a1^2-96*a1*a2*a3+64*a1*a3^3+25*a2^3+69*a2^2*a3^2-117\
   *a2*a3^4+39*a3^6)
==>             _[3]=(a2-a3^2)
==>       [3]:
==>          [1]:
==>             _[1]=1
==>          [2]:
==>             _[1]=(16*a1^2-96*a1*a2*a3+64*a1*a3^3+25*a2^3+69*a2^2*a3^2-117\
   *a2*a3^4+39*a3^6)
==>             _[2]=(a0-4*a1*a3+6*a2*a3^2-3*a3^4)
==>          [3]:
==>             _[1]=(a1-3*a2*a3+2*a3^3)
==>             _[2]=(a2-a3^2)
==>       [4]:
==>          [1]:
==>             _[1]=(a1^2-2*a1*a3^3+a3^6)
==>          [2]:
==>             _[1]=(a2-a3^2)
==>             _[2]=(a0-4*a1*a3+3*a3^4)
==>          [3]:
==>             _[1]=(a1-3*a2*a3+2*a3^3)
==>       [5]:
==>          [1]:
==>             _[1]=1
==>          [2]:
==>             _[1]=(a1-3*a2*a3+2*a3^3)
==>             _[2]=(a0-6*a2*a3^2+5*a3^4)
==>          [3]:
==>             _[1]=(a2-a3^2)
==> [2]:
==>    [1]:
==>       _[1]=x3
==>       _[2]=x2^2
==>       _[3]=x1^3
==>    [2]:
==>       [1]:
==>          [1]:
==>             _[1]=x3+(a3)
==>             _[2]=x2^2+(2*a3)*x2+(a3^2)
==>             _[3]=x1^3+(3*a3)*x1^2+(3*a3^2)*x1+(a3^3)
==>          [2]:
==>             _[1]=(a2-a3^2)
==>             _[2]=(a1-a3^3)
==>             _[3]=(a0-a3^4)
==>          [3]:
==>             _[1]=1


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