|  |  D.2.4.1 grobcov Procedure from librarygrobcov.lib(see  grobcov_lib).
 
Example:Usage:
grobcov(ideal F[,options]);
F: ideal in Q[a][x] (a=parameters, x=variables) to be
 discussed.This is the fundamental routine of the
 library. It computes the Groebner Cover of a parametric
ideal F in Q[a][x]. See
 A. Montes , M. Wibmer, "Groebner Bases for Polynomial
 Systems with parameters".
 JSC 45 (2010) 1391-1425.)
 or the not yet published book
 A. Montes. " The Groebner Cover" (Discussing
 Parametric Polynomial Systems).
 The Groebner Cover of a parametric ideal F 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 reducedGroebner 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).
 
Return:
The list [[lpp_1,basis_1,segment_1], ...,
[lpp_s,basis_s,segment_s]]
 optionally [[ 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.
 With option ("showhom",1) the lpph will be
 shown: 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.
 
Options:
An option is a pair of arguments: string,
integer. To modify the default options, pairs
 of arguments -option name, value- of valid options
 must be added to the call.
 "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.
 "showhom",0-1: The default is ("showhom",0).
 Setting "showhom",1 will output the set
 of lpp of the homogenized ideal of each segment
 as last element. One can give none or whatever
 of these options.
 
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.
 
 |  | LIB "grobcov.lib";
// Casas conjecture for degree 4:
if(defined(R)){kill R;}
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)
==> [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
// EXAMPLE:
// M. Rychlik robot;
// Complexity and Applications of Parametric Algorithms of
//    Computational Algebraic Geometry.;
// In: Dynamics of Algorithms, R. de la Llave, L. Petzold and J. Lorenz eds.;
// IMA Volumes in Mathematics and its Applications,
//    Springer-Verlag 118: 1-29 (2000).;
//    (18. Mathematical robotics: Problem 4, two-arm robot)."
if (defined(R)){kill R;}
ring R=(0,a,b,l2,l3),(c3,s3,c1,s1), dp;
short=0;
ideal S12=a-l3*c3-l2*c1,b-l3*s3-l2*s1,c1^2+s1^2-1,c3^2+s3^2-1;
S12;
==> S12[1]=(-l3)*c3+(-l2)*c1+(a)
==> S12[2]=(-l3)*s3+(-l2)*s1+(b)
==> S12[3]=c1^2+s1^2-1
==> S12[4]=c3^2+s3^2-1
grobcov(S12);
==> [1]:
==>    [1]:
==>       _[1]=c1
==>       _[2]=s3
==>       _[3]=c3
==>       _[4]=s1^2
==>    [2]:
==>       _[1]=(2*a*l2)*c1+(2*b*l2)*s1+(-a^2-b^2-l2^2+l3^2)
==>       _[2]=(l3)*s3+(l2)*s1+(-b)
==>       _[3]=(2*a*l3)*c3+(-2*b*l2)*s1+(-a^2+b^2+l2^2-l3^2)
==>       _[4]=(4*a^2*l2^2+4*b^2*l2^2)*s1^2+(-4*a^2*b*l2-4*b^3*l2-4*b*l2^3+4*\
   b*l2*l3^2)*s1+(a^4+2*a^2*b^2-2*a^2*l2^2-2*a^2*l3^2+b^4+2*b^2*l2^2-2*b^2*l\
   3^2+l2^4-2*l2^2*l3^2+l3^4)
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=0
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>             [2]:
==>                _[1]=(l2)
==>             [3]:
==>                _[1]=(a^2+b^2)
==>             [4]:
==>                _[1]=(a)
==> [2]:
==>    [1]:
==>       _[1]=s1
==>       _[2]=s3
==>       _[3]=c3
==>       _[4]=c1^2
==>    [2]:
==>       _[1]=(2*b*l2)*s1+(-b^2-l2^2+l3^2)
==>       _[2]=(2*b*l3)*s3+(-b^2+l2^2-l3^2)
==>       _[3]=(l3)*c3+(l2)*c1
==>       _[4]=(4*b^2*l2^2)*c1^2+(b^4-2*b^2*l2^2-2*b^2*l3^2+l2^4-2*l2^2*l3^2+\
   l3^4)
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(a)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(a)
==>             [2]:
==>                _[1]=(l2)
==>                _[2]=(a)
==>             [3]:
==>                _[1]=(b)
==>                _[2]=(a)
==> [3]:
==>    [1]:
==>       _[1]=1
==>    [2]:
==>       _[1]=1
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(b)
==>             _[2]=(a)
==>          [2]:
==>             [1]:
==>                _[1]=(l2+l3)
==>                _[2]=(b)
==>                _[3]=(a)
==>             [2]:
==>                _[1]=(l3)
==>                _[2]=(b)
==>                _[3]=(a)
==>             [3]:
==>                _[1]=(l2-l3)
==>                _[2]=(b)
==>                _[3]=(a)
==>       [2]:
==>          [1]:
==>             _[1]=(l2)
==>          [2]:
==>             [1]:
==>                _[1]=(l2)
==>                _[2]=(a^2+b^2-l3^2)
==>             [2]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==> [4]:
==>    [1]:
==>       _[1]=s3
==>       _[2]=c3
==>       _[3]=c1^2
==>    [2]:
==>       _[1]=(l2^2*l3+2*l2^2-l3^3)*s3+(2*l2*l3)*s1+(b*l3^2)
==>       _[2]=(l2^2*l3+2*l2^2-l3^3)*c3+(2*l2*l3)*c1+(a*l3^2)
==>       _[3]=c1^2+s1^2-1
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(l2-l3)
==>             _[2]=(b)
==>             _[3]=(a)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==>                _[3]=(b)
==>                _[4]=(a)
==>       [2]:
==>          [1]:
==>             _[1]=(l2+l3)
==>             _[2]=(b)
==>             _[3]=(a)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==>                _[3]=(b)
==>                _[4]=(a)
==>       [3]:
==>          [1]:
==>             _[1]=(l2)
==>             _[2]=(a^2+b^2-l3^2)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==>                _[3]=(a^2+b^2)
==> [5]:
==>    [1]:
==>       _[1]=c1^2
==>       _[2]=c3^2
==>    [2]:
==>       _[1]=c1^2+s1^2-1
==>       _[2]=c3^2+s3^2-1
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(l3)
==>             _[2]=(l2)
==>             _[3]=(b)
==>             _[4]=(a)
==>          [2]:
==>             [1]:
==>                _[1]=1
==> [6]:
==>    [1]:
==>       _[1]=1
==>    [2]:
==>       _[1]=1
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(l3)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(a^2+b^2-l2^2)
==>             [2]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==> [7]:
==>    [1]:
==>       _[1]=1
==>    [2]:
==>       _[1]=1
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(l3)
==>             _[2]=(l2)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==>                _[3]=(a^2+b^2)
==> [8]:
==>    [1]:
==>       _[1]=s1
==>       _[2]=c1
==>       _[3]=c3^2
==>    [2]:
==>       _[1]=(l2)*s1+(-b)
==>       _[2]=(l2)*c1+(-a)
==>       _[3]=c3^2+s3^2-1
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(l3)
==>             _[2]=(a^2+b^2-l2^2)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==>                _[3]=(a^2+b^2)
==> [9]:
==>    [1]:
==>       _[1]=1
==>    [2]:
==>       _[1]=1
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(l3)
==>             _[2]=(l2)
==>             _[3]=(a^2+b^2)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==>                _[3]=(b)
==>                _[4]=(a)
==> [10]:
==>    [1]:
==>       _[1]=s1
==>       _[2]=c1
==>       _[3]=s3
==>       _[4]=c3
==>    [2]:
==>       _[1]=(4*b*l2^3-4*b*l2*l3^2)*s1+(-4*b^2*l2^2-l2^4+2*l2^2*l3^2-l3^4)
==>       _[2]=(4*b^2*l2^3-4*b^2*l2*l3^2)*c1+(-4*a*b^2*l2^2+a*l2^4-2*a*l2^2*l\
   3^2+a*l3^4)
==>       _[3]=(4*b*l2^2*l3-4*b*l3^3)*s3+(4*b^2*l3^2+l2^4-2*l2^2*l3^2+l3^4)
==>       _[4]=(4*b^2*l2^2*l3-4*b^2*l3^3)*c3+(4*a*b^2*l3^2-a*l2^4+2*a*l2^2*l3\
   ^2-a*l3^4)
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(a^2+b^2)
==>          [2]:
==>             [1]:
==>                _[1]=(l2+l3)
==>                _[2]=(a^2+b^2)
==>             [2]:
==>                _[1]=(l3)
==>                _[2]=(a^2+b^2)
==>             [3]:
==>                _[1]=(l2-l3)
==>                _[2]=(a^2+b^2)
==>             [4]:
==>                _[1]=(l2)
==>                _[2]=(a^2+b^2)
==>             [5]:
==>                _[1]=(b)
==>                _[2]=(a)
==> [11]:
==>    [1]:
==>       _[1]=1
==>    [2]:
==>       _[1]=1
==>    [3]:
==>       [1]:
==>          [1]:
==>             _[1]=(l2-l3)
==>             _[2]=(a^2+b^2)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==>                _[3]=(a^2+b^2)
==>             [2]:
==>                _[1]=(l2-l3)
==>                _[2]=(b)
==>                _[3]=(a)
==>       [2]:
==>          [1]:
==>             _[1]=(l2+l3)
==>             _[2]=(a^2+b^2)
==>          [2]:
==>             [1]:
==>                _[1]=(l3)
==>                _[2]=(l2)
==>                _[3]=(a^2+b^2)
==>             [2]:
==>                _[1]=(l2+l3)
==>                _[2]=(b)
==>                _[3]=(a)
 | 
 
 |