SINGULAR                                 /  Development
 A Computer Algebra System for Polynomial Computations       /   version 4.0.3
                                                           0<
 by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Jan 2016
FB Mathematik der Universitaet, D-67653 Kaiserslautern        \
> //---[CLO, p. 309]
. //---Specialization of Groebner bases
. ring r=0,(c1,s1,c2,s2,a,b,l2,l3),(dp(4),dp); //product order
> option(redSB);
>
. ideal id=l3*(c1*c2-s1*s2)+l2*c1-a,
. l3*(c1*s2+c2*s1)+l2*s1-b,
. c1^2+s1^2-1,c2^2+s2^2-1;
>
. ideal j = std(id);
> j;
j[1]=2*c2*l2*l3-a^2-b^2+l2^2+l3^2
j[2]=s1*a^2+s1*b^2-c2*b*l3+s2*a*l3-b*l2
j[3]=c1*b-s1*a-s2*l3
j[4]=c1*a+s1*b-c2*l3-l2
j[5]=2*s2^2*l2*l3+c2*a^2+c2*b^2-c2*l2^2-c2*l3^2-2*l2*l3
j[6]=2*s1*s2*l2*l3-c1*l2^2+c1*l3^2-c2*a*l3-s2*b*l3+a*l2
j[7]=2*c1*s2*l2*l3+s1*l2^2-s1*l3^2+c2*b*l3-s2*a*l3-b*l2
j[8]=c2^2+s2^2-1
j[9]=s1*c2*l3+c1*s2*l3+s1*l2-b
j[10]=s1*c2*l2-c1*s2*l2+s1*l3-c2*b+s2*a
j[11]=c1*c2*l3-s1*s2*l3+c1*l2-a
j[12]=c1*c2*l2+s1*s2*l2+c1*l3-c2*a-s2*b
j[13]=c1^2+s1^2-1
> //---[CLO, p. 309, line 16] should be corrected as follows:
. // 2*l_2l_3, a^2+b^2, b, a, 2*l_2l_3, 2*l_2l_3, 2*l_2l_3, 1, l_3,
. // l_2, l_3, l_2, 1.
. //---Groebner cover di <x+y,x+t*y> in Q(t)[x,y]
. LIB "grobcov.lib";
ideal I = x+y, x+t*y;
grobcov(I);// ** loaded /usr/bin/../share/singular/LIB/grobcov.lib (4.0.2.0,Jul_2015)
// ** loaded /usr/bin/../share/singular/LIB/qhmoduli.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/rinvar.lib (4.0.1.1,Jan_2015)
// ** loaded /usr/bin/../share/singular/LIB/ring.lib (4.0.2.2,Jan_2016)
// ** loaded /usr/bin/../share/singular/LIB/general.lib (4.0.0.1,Jan_2014)
// ** loaded /usr/bin/../share/singular/LIB/matrix.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/nctools.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/random.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/poly.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/inout.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/zeroset.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/primitiv.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/elim.lib (4.0.0.1,Jan_2014)
// ** loaded /usr/bin/../share/singular/LIB/presolve.lib (4.0.2.2,Jan_2016)
// ** loaded /usr/bin/../share/singular/LIB/primdec.lib (4.0.2.0,Apr_2015)
// ** loaded /usr/bin/../share/singular/LIB/absfact.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/triang.lib (4.0.0.0,Jun_2013)
> ring rr = (0,t),(x,y),lp;
> ideal I = x+y, x+t*y;
> grobcov(I);
[1]:
   [1]:
      _[1]=y
      _[2]=x
   [2]:
      _[1]=y
      _[2]=x
   [3]:
      [1]:
         [1]:
            _[1]=0
         [2]:
            [1]:
               _[1]=(t-1)
   [4]:
      y,x
[2]:
   [1]:
      _[1]=x
   [2]:
      _[1]=x+y
   [3]:
      [1]:
         [1]:
            _[1]=(t-1)
         [2]:
            [1]:
               _[1]=1
   [4]:
      x
> //---Torniamo al robot 1
. ring r=(0,l3,l2,a,b),(c1,s1,c2,s2),dp;
// ** redefining r **
> option(redSB);
> ideal id=l3*(c1*c2-s1*s2)+l2*c1-a,
. l3*(c1*s2+c2*s1)+l2*s1-b,
. c1^2+s1^2-1,c2^2+s2^2-1;
> list g = grobcov(id);
> g;
[1]:
   [1]:
      _[1]=c2
      _[2]=s1
      _[3]=c1
      _[4]=s2^2
   [2]:
      _[1]=(2*l3*l2)*c2+(l3^2+l2^2-a^2-b^2)
      _[2]=(2*l2*a^2+2*l2*b^2)*s1+(2*l3*l2*a)*s2+(l3^2*b-l2^2*b-a^2*b-b^3)
      _[3]=(2*l2*a^2+2*l2*b^2)*c1+(-2*l3*l2*b)*s2+(l3^2*a-l2^2*a-a^3-a*b^2)
      _[4]=(4*l3^2*l2^2)*s2^2+(l3^4-2*l3^2*l2^2-2*l3^2*a^2-2*l3^2*b^2+l2^4-2*l2^2*a^2-2*l2^2*b^2+a^4+2*a^2*b^2+b^4)
   [3]:
      [1]:
         [1]:
            _[1]=0
         [2]:
            [1]:
               _[1]=(a^2+b^2)
            [2]:
               _[1]=(l2)
            [3]:
               _[1]=(l3)
   [4]:
      c2,s1,c1,s2^2
[2]:
   [1]:
      _[1]=1
   [2]:
      _[1]=1
   [3]:
      [1]:
         [1]:
            _[1]=(l3)
         [2]:
            [1]:
               _[1]=(l2^2-a^2-b^2)
               _[2]=(l3)
            [2]:
               _[1]=(a^2+b^2)
               _[2]=(l3)
      [2]:
         [1]:
            _[1]=(l2)
         [2]:
            [1]:
               _[1]=(l2)
               _[2]=(l3^2-a^2-b^2)
            [2]:
               _[1]=(a^2+b^2)
               _[2]=(l2)
   [4]:
      @t,s1,c1,c2^2
[3]:
   [1]:
      _[1]=s1
      _[2]=c1
      _[3]=c2^2
   [2]:
      _[1]=(l2^2*a^2+l2^2*b^2+l2*a^2+l2*b^2-a^4-2*a^2*b^2-b^4)*s1+(l3*a^2*b+l3*b^3)*c2+(-l3*a^3-l3*a*b^2)*s2+(-l2^2*b)
      _[2]=(l2^2*a^2+l2^2*b^2+l2*a^2+l2*b^2-a^4-2*a^2*b^2-b^4)*c1+(l3*a^3+l3*a*b^2)*c2+(l3*a^2*b+l3*b^3)*s2+(-l2^2*a)
      _[3]=c2^2+s2^2-1
   [3]:
      [1]:
         [1]:
            _[1]=(l2^2-a^2-b^2)
            _[2]=(l3)
         [2]:
            [1]:
               _[1]=(a^2+b^2)
               _[2]=(l2)
               _[3]=(l3)
      [2]:
         [1]:
            _[1]=(l2)
            _[2]=(l3^2-a^2-b^2)
         [2]:
            [1]:
               _[1]=(a^2+b^2)
               _[2]=(l2)
               _[3]=(l3)
   [4]:
      s1,c1,c2^2
[4]:
   [1]:
      _[1]=1
   [2]:
      _[1]=1
   [3]:
      [1]:
         [1]:
            _[1]=(a^2+b^2)
            _[2]=(l2)
            _[3]=(l3)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l2)
               _[4]=(l3)
   [4]:
      @t^2,c2*@t,c1,c2^2
[5]:
   [1]:
      _[1]=c2^2
      _[2]=c1^2
   [2]:
      _[1]=c2^2+s2^2-1
      _[2]=c1^2+s1^2-1
   [3]:
      [1]:
         [1]:
            _[1]=(b)
            _[2]=(a)
            _[3]=(l2)
            _[4]=(l3)
         [2]:
            [1]:
               _[1]=1
   [4]:
      c2^2,c1^2
[6]:
   [1]:
      _[1]=1
   [2]:
      _[1]=1
   [3]:
      [1]:
         [1]:
            _[1]=(a^2+b^2)
            _[2]=(l3)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l3)
            [2]:
               _[1]=(a^2+b^2)
               _[2]=(l2)
               _[3]=(l3)
   [4]:
      @t,c1,c2^2,s1*c2
[7]:
   [1]:
      _[1]=1
   [2]:
      _[1]=1
   [3]:
      [1]:
         [1]:
            _[1]=(b)
            _[2]=(a)
            _[3]=(l3)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l2)
               _[4]=(l3)
   [4]:
      @t,c2^2,s1*c2,c1*c2,c1^2
[8]:
   [1]:
      _[1]=1
   [2]:
      _[1]=1
   [3]:
      [1]:
         [1]:
            _[1]=(a^2+b^2)
            _[2]=(l2)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l2)
            [2]:
               _[1]=(a^2+b^2)
               _[2]=(l2)
               _[3]=(l3)
   [4]:
      @t,c2,c1,s1*s2
[9]:
   [1]:
      _[1]=1
   [2]:
      _[1]=1
   [3]:
      [1]:
         [1]:
            _[1]=(b)
            _[2]=(a)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l3+l2)
            [2]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l3)
            [3]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l3-l2)
   [4]:
      @t,s2,c2,c1^2
[10]:
   [1]:
      _[1]=s2
      _[2]=c2
      _[3]=s1
      _[4]=c1
   [2]:
      _[1]=(2*l3*l2*b)*s2+(-l3^2*a+l2^2*a)
      _[2]=(2*l3*l2)*c2+(l3^2+l2^2)
      _[3]=(4*l3^2*l2*b-4*l2^3*b)*s1+(l3^4-2*l3^2*l2^2+l2^4+4*l2^2*b^2)
      _[4]=(4*l3^2*l2*b^2-4*l2^3*b^2)*c1+(-l3^4*a+2*l3^2*l2^2*a-l2^4*a+4*l2^2*a*b^2)
   [3]:
      [1]:
         [1]:
            _[1]=(a^2+b^2)
         [2]:
            [1]:
               _[1]=(a^2+b^2)
               _[2]=(l2)
            [2]:
               _[1]=(b)
               _[2]=(a)
            [3]:
               _[1]=(a^2+b^2)
               _[2]=(l3+l2)
            [4]:
               _[1]=(a^2+b^2)
               _[2]=(l3)
            [5]:
               _[1]=(a^2+b^2)
               _[2]=(l3-l2)
   [4]:
      s2,c2,s1*@t,c1
[11]:
   [1]:
      _[1]=s2
      _[2]=c2
      _[3]=c1^2
   [2]:
      _[1]=s2
      _[2]=(l3)*c2+(l2)
      _[3]=c1^2+s1^2-1
   [3]:
      [1]:
         [1]:
            _[1]=(b)
            _[2]=(a)
            _[3]=(l3-l2)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l2)
               _[4]=(l3)
      [2]:
         [1]:
            _[1]=(b)
            _[2]=(a)
            _[3]=(l3+l2)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l2)
               _[4]=(l3)
   [4]:
      s2,c2,c1^2
[12]:
   [1]:
      _[1]=1
   [2]:
      _[1]=1
   [3]:
      [1]:
         [1]:
            _[1]=(a^2+b^2)
            _[2]=(l3-l2)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l3-l2)
            [2]:
               _[1]=(a^2+b^2)
               _[2]=(l2)
               _[3]=(l3)
      [2]:
         [1]:
            _[1]=(a^2+b^2)
            _[2]=(l3+l2)
         [2]:
            [1]:
               _[1]=(b)
               _[2]=(a)
               _[3]=(l3+l2)
            [2]:
               _[1]=(a^2+b^2)
               _[2]=(l2)
               _[3]=(l3)
   [4]:
      @t^2,s2,c2,c1
>