15.1.1 Other methods

For a plane curve, you can use Singular's closed_points command. The input is the vanishing ideal

of the curve

in a ring of

variables

. The closed_points command returns a list of prime ideals (each a Gröbner basis), corresponding to the (distinct affine closed) points of

. Here's an example:

sage: singular_console()
                     SINGULAR                             /  Development
 A Computer Algebra System for Polynomial Computations   /   version 3-0-1
                                                       0<
     by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   October 2005
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
// ** executing /home/wdj/sagefiles/sage-0.9.4/local/LIB/.singularrc
> LIB "brnoeth.lib";
> ring s = 2,(x,y),lp;
> ideal I = x4+x,y4+y;
> list L = closed_points(I);
> L;
[1]:
   _[1] = y
   _[2] = x
[2]:
   _[1] = y
   _[2] = x+1
[3]:
   _[1] = y
   _[2] = x2+x+1
[4]:
   _[1] = y+1
   _[2] = x
[5]:
   _[1] = y+1
   _[2] = x+1
[6]:
   _[1] = y+1
   _[2] = x2+x+1
[7]:
   _[1] = y2+y+1
   _[2] = x+1
[8]:
   _[1] = y2+y+1
   _[2] = x
[9]:
   _[1] = y2+y+1
   _[2] = x+y
[10]:
   _[1] = y2+y+1
   _[2] = x+y+1
> Auf Wiedersehen.

sage: singular.lib("brnoeth.lib")
sage: s = singular.ring(2,'(x,y)','lp')
sage: I = singular.ideal('[x^4+x, y^4+y]')
sage: L = singular.closed_points(I)
sage: # Here you have all the points :
sage: print L
[1]:
   _[1]=y^2+y+1
   _[2]=x+1
...

Another way to compute rational points is to use Singular's NSplaces command. Here's the Klein quartic over

done this way:

sage: singular.LIB("brnoeth.lib")
sage: s = singular.ring(2,'(x,y)','lp')
...
sage: f = singular.poly('x3y+y3+x')
...
sage: klein1 = f.Adj_div(); print klein1
[1]:
   [1]:
      //   characteristic : 2
//   number of vars : 2
//        block   1 : ordering lp
//                  : names    x y
//        block   2 : ordering C
...
sage: # define a curve X = {f = 0} over GF(2)
sage: klein2 = singular.NSplaces(3,klein1)
sage: print singular.eval('extcurve(3,%s)'%klein2.name())
Total number of rational places : NrRatPl = 23
...
sage: klein3 = singular.extcurve(3, klein2)

Above we defined a curve $X = \{f = 0\}$ over in Singular.

sage: print klein1
[1]:
   [1]:
      //   characteristic : 2
//   number of vars : 2
//        block   1 : ordering lp
//                  : names    x y
//        block   2 : ordering C
   [2]:
      //   characteristic : 2
//   number of vars : 3
//        block   1 : ordering lp
//                  : names    x y z
//        block   2 : ordering C
[2]:
   4,3
[3]:
   [1]:
      1,1
   [2]:
      1,2
[4]:
   0
[5]:
   [1]:
      [1]:
         //   characteristic : 2
//   number of vars : 3
//        block   1 : ordering ls
//                  : names    x y t
//        block   2 : ordering C
      [2]:
         1,1
sage: print klein1[3]
[1]:
   1,1
[2]:
   1,2

For the places of degree :

sage: print klein2[3]
[1]:
   1,1
[2]:
   1,2
[3]:
   3,1
[4]:
   3,2
[5]:
   3,3
[6]:
   3,4
[7]:
   3,5
[8]:
   3,6
[9]:
   3,7

Each point below is a pair: (degree, point index number).

sage: print klein3[3]
[1]:
   1,1
[2]:
   1,2
[3]:
   3,1
[4]:
   3,2
[5]:
   3,3
[6]:
   3,4
[7]:
   3,5
[8]:
   3,6
[9]:
   3,7

To actually get the points of :

sage: R = klein3[1][5]
sage: R.set_ring()
sage: singular("POINTS")
[1]:
   [1]:
      0
   [2]:
      1
   [3]:
      0
[2]:
   [1]:
      1
   [2]:
      0
   [3]:
      0
[3]:
   [1]:
      (a^2+a)
   [2]:
      (a)
   [3]:
      1
[4]:
   [1]:
      (a+1)
   [2]:
      (a^2)
   [3]:
      1
...

plus 19 others (omitted). There are a total of rational points. Here represents a primitive element in .