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
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i4 : i = pfaffians(4,m)
o4 = ideal(x x - x x + x x )
2 3 1 4 0 5
o4 : Ideal of R
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i5 : p = randomPoint i
o5 = ideal (x - 11x , x - 9x , x - 18x , x + 11x , - 4x + x )
3 4 2 4 1 4 0 4 4 5
o5 : Ideal of R
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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 , c , c , c , o , o , o , o , o , o , o , o , o , o , o , o , x , x , y , y , 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 1,4 1,5 1,6 2,1 2,2 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 1 2 3
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