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D.2.4.1 grobcov
Procedure from library grobcov.lib (see grobcov_lib).
- Return:
- The list
(
(lpp_1,basis_1,P-representation_1),
...
(lpp_s,basis_s,P-represntation_s)
)
The lpp are constant over a segment and correspond to the
set of lpp of the reduced Groebner basis for each point
of the segment.
Basis: to each element of lpp corresponds an I-regular function given Groebner basis, and it is given in full representation (by
in full representation (by default option "ext",1) or in
generic representation (option "ext",0). The regular function is
the corresponding element of the reduced Groebner basis for
each point of the segment with the given lpp.
For each point in the segment, the polynomial or the set of
polynomials representing it, if they do not specialize to 0,
then after normalization, specialize to the corresponding
element of the reduced Groebner basis.
The P-representation of a segment is of the form
((p_1,(p_11,..,p_1k1)),..,(p_r,(p_r1,..,p_rkr))
representing the segment U_i (V(p_i) \ U_j (V(p_ij))), where the
p's are prime ideals.
- Note:
- The basering R, must be of the form Q[a][x], a=parameters,
x=variables, and should be defined previously. The ideal must
be defined on R.
parametric ideal, multigrobcov, gencase1.
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);
grobcov(F);
==> [1]:
==> [1]:
==> _[1]=1
==> [2]:
==> _[1]=1
==> [3]:
==> [1]:
==> [1]:
==> _[1]=0
==> [2]:
==> [1]:
==> _[1]=(a2-a3^2)
==> _[2]=(a1-a3^3)
==> _[3]=(a0-a3^4)
==> [2]:
==> [1]:
==> _[1]=x3
==> _[2]=x2^2
==> _[3]=x1^3
==> [2]:
==> _[1]=x3+(a3)
==> _[2]=x2^2+(2*a3)*x2+(a3^2)
==> _[3]=x1^3+(3*a3)*x1^2+(3*a3^2)*x1+(a3^3)
==> [3]:
==> [1]:
==> [1]:
==> _[1]=(a2-a3^2)
==> _[2]=(a1-a3^3)
==> _[3]=(a0-a3^4)
==> [2]:
==> [1]:
==> _[1]=1
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