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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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