Singular

D.2.4.1 grobcov

Usage:

grobcov(F); This is the fundamental routine of the
library. It computes the Groebner cover of a parametric ideal. See Montes A., Wibmer M.,
"Groebner Bases for Polynomial Systems with parameters". JSC 45 (2010) 1391-1425.)
The Groebner Cover of a parametric ideal consist of a set of pairs(S_i,B_i), where the S_i are disjoint locally closed segments of the parameter space, and the B_i are the reduced Groebner bases of the ideal on every point of S_i.

The ideal F must be defined on a parametric ring Q[a][x]. (a=parameters, x=variables)
Options: To modify the default options, pair of arguments -option name, value- of valid options must be added to the call.

Options:
"null",ideal E: The default is ("null",ideal(0)).
"nonnull",ideal N: The default is ("nonnull",ideal(1)). When options "null" and/or "nonnull" are given, then
the parameter space is restricted to V(E) \ V(N).
"can",0-1: The default is ("can",1). With the default option the homogenized ideal is computed before obtaining the Groebner Cover, so that the result is the canonical
Groebner Cover. Setting ("can",0) only homogenizes the basis so the result is not exactly canonical, but the
computation is shorter.
"ext",0-1: The default is ("ext",0). With the default
("ext",0), only the generic representation of the bases is computed (single polynomials, but not specializing to non-zero for every point of the segment. With option ("ext",1) the full representation of the bases is computed (possible sheaves) and sometimes a simpler result is obtained,
but the computation is more time consuming.
"rep",0-1-2: The default is ("rep",0) and then the segments are given in canonical P-representation. Option ("rep",1) represents the segments in canonical C-representation, and option ("rep",2) gives both representations.
"comment",0-3: The default is ("comment",0). Setting
"comment" higher will provide information about the
development of the computation.
One can give none or whatever of these options.

Return:

The list ( (lpp_1,basis_1,segment_1,lpph_1), ... (lpp_s,basis_s,segment_s,lpph_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.
The lpph corresponds to the lpp of the homogenized ideal and is different for each segment. It is given as a string, and shown only for information. With the default option "can",1, the segments have different lpph.

Basis: to each element of lpp corresponds an I-regular function given in full representation (by option ("ext",1)) or in generic representation (default option ("ext",0)). The I-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, specializes to the corresponding element of the reduced Groebner basis. In the full representation at least one of the polynomials representing the I-regular function specializes to non-zero.

With the default option ("rep",0) the representation of the segment is the P-representation.
With option ("rep",1) the representation of the segment is the C-representation.
With option ("rep",2) both representations of the segment are given.

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 Union_i ( V(p_i) \ ( Union_j V(p_ij) ) ), where the p's are prime ideals.

The C-representation of a segment is of the form
(E,N) representing V(E) \ V(N), and the ideals E and N are radical and N contains E.

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.

LIB "grobcov.lib";
// Casas conjecture for degree 4:
ring R=(0,a0,a1,a2,a3,a4),(x1,x2,x3),dp;
short=0;
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)
==>    [4]:
==>       1
==> [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
==>    [4]:
==>       x3,x2^2,x1^3