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NumericalGodeaux :: randomPoint

randomPoint -- compute a rational point in a variety

Synopsis

Description

Given an ideal in a polynomial ring over a finite field, this procedure computes, if possible, a rational point in the corresponding variety. We proceed by intersecting with random hyperplanes down to a set of points and repeat this until we find a rational point (if existing). If an additional polynomial ring is indicated, then we compute a point in this ring, if possible. If the ideal $I$ is just the complete intersection of the Pfaffian relations from our construction, then we can use the corresponding skew-symmetric matrices from the list Ms to cut down to some linear subspaces which may speed up the computation. If the ideal $I$ defines an empty vanishing locus, an error message is printed.

i1 : kk = ZZ/37;
 
i2 : R = kk[x_0..x_5]

o2 = R

o2 : PolynomialRing
 
i3 : m = genericSkewMatrix(R,4)

o3 = | 0    x_0  x_1  x_2 |
     | -x_0 0    x_3  x_4 |
     | -x_1 -x_3 0    x_5 |
     | -x_2 -x_4 -x_5 0   |

             4       4
o3 : Matrix R  <--- R
 
i4 : i = pfaffians(4,m)

o4 = ideal(x x  - x x  + x x )
            2 3    1 4    0 5

o4 : Ideal of R
 
i5 : p = randomPoint i

o5 = ideal (x  + 9x , x  - 14x , x  + 4x , x  - 8x , x  + 16x )
             4     5   3      5   2     5   1     5   0      5

o5 : Ideal of R
 
i6 : s = "1111";
 
i7 : (relLin,relPfaf,d1',d2,Ms) = setupGodeaux(kk,s);
 
i8 : p = randomPoint(ideal relPfaf,Ms)

o8 = ideal (a     , a      + 17a     , a      + 15a     , a      + 7a     ,
             0,0,2   0,0,3      0,0,1   1,0,1      0,0,1   1,1,2     0,0,1 
     ------------------------------------------------------------------------
     a      - 7a     , a      - 10a     , a      + 8a     , a      - 9a     ,
      1,1,3     0,0,1   2,0,2      0,0,1   2,1,2     0,0,1   2,2,3     0,0,1 
     ------------------------------------------------------------------------
     a      - 4a     , a      + 15a     , a      - 4a     )
      3,0,3     0,0,1   3,1,3      0,0,1   3,2,3     0,0,1

o8 : Ideal of kk[a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , c   , c   , c   , c   ..c   , c   , c   ..c   , c   ..c   , c   , c   , c   ..c   , c   , c   , c   , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , x ..x , y ..y ]
                  3,2,3   3,1,3   3,0,3   2,2,3   2,1,2   2,0,2   1,1,3   1,1,2   1,0,1   0,0,3   0,0,2   0,0,1   0,0   0,2   0,4   0,6   0,7   1,1   1,3   1,6   2,1   2,3   2,5   2,7   3,0   3,1   3,3   3,5   3,7   1,0,0   2,0,1   2,1,2   3,0,0   3,1,0   4,0,1   4,2,1   4,3,3   5,1,2   5,2,2   5,3,3   5,4,3   0   1   0   3

Ways to use randomPoint :

For the programmer

The object randomPoint is a method function.