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randomPointsOnRationalVariety(Ideal,ZZ) -- find random points on a variety that can be detected to be rational

Synopsis

Description

i1 : kk = ZZ/101;
i2 : S = kk[a..f];
i3 : I = minors(2, genericSymmetricMatrix(S, 3))

               2                                                  2         
o3 = ideal (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, -
     ------------------------------------------------------------------------
                                             2
     c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)

o3 : Ideal of S
i4 : pts = randomPointsOnRationalVariety(I, 4)

o4 = {| 1 49 24 -23 -36 -30 |, | 23 -29 -29 19 19 19 |, | 38 -11 -10 -42 -29
     ------------------------------------------------------------------------
     -8 |, | -37 -35 -22 -14 -29 -24 |}

o4 : List
i5 : for p in pts list sub(I, p) == 0

o5 = {true, true, true, true}

o5 : List
i6 : S = kk[a..d];
i7 : F = groebnerFamily ideal"a2,ab,ac,b2"

             2                      2                      2               
o7 = ideal (a  + t b*c + t a*d + t c  + t b*d + t c*d + t d , a*b + t b*c +
                  1       3       2      4       5       6           7     
     ------------------------------------------------------------------------
                2                         2                              2  
     t a*d + t c  + t  b*d + t  c*d + t  d , a*c + t  b*c + t  a*d + t  c  +
      9       8      10       11       12           13       15       14    
     ------------------------------------------------------------------------
                           2   2                         2                  
     t  b*d + t  c*d + t  d , b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d
      16       17       18          19       21       20      22       23   
     ------------------------------------------------------------------------
           2
     + t  d )
        24

o7 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  , t  , t , t , t  , t  , t  , t  , t  , t  , t  , t  ][a..d]
                  6   5   12   2   4   11   18   24   1   3   8   10   17   23   7   9   14   16   20   22   13   15   19   21
i8 : J = groebnerStratum F;

o8 : Ideal of kk[t , t , t  , t , t , t  , t  , t  , t , t , t , t  , t  , t  , t , t , t  , t  , t  , t  , t  , t  , t  , t  ]
                  6   5   12   2   4   11   18   24   1   3   8   10   17   23   7   9   14   16   20   22   13   15   19   21
i9 : compsJ = decompose J;
i10 : compsJ = compsJ/trim;
i11 : #compsJ == 2

o11 = true
i12 : compsJ/dim

o12 = {11, 8}

o12 : List

There are 2 components. We attempt to find points on each of these two components. We are successful. This indicates that the corresponding varieties are both rational. Also, if we can find one point, we can find as many as we want.

i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)

      +-----------------------------------------------------------------------------------------+
o13 = || 42 -50 -50 38 -39 6 -1 47 49 -18 18 -28 -47 19 48 34 -13 11 -16 -38 39 21 -43 -39 |    |
      +-----------------------------------------------------------------------------------------+
      || -44 -31 -9 -21 -42 -36 -47 -20 38 -34 -3 -43 22 16 -35 2 -48 32 -28 -15 -47 38 -47 45 ||
      +-----------------------------------------------------------------------------------------+
      || 45 -29 -1 -4 42 -35 4 -13 18 -17 1 21 39 -23 50 15 -11 -11 19 47 -16 7 48 43 |         |
      +-----------------------------------------------------------------------------------------+
      || 35 -44 -33 -8 21 -2 -44 -20 19 -28 19 27 11 40 34 33 1 -14 35 36 11 -38 -3 46 |        |
      +-----------------------------------------------------------------------------------------+
      || 33 47 46 16 -22 -25 -44 -36 -30 -37 30 -25 -47 29 -41 2 -13 -41 -47 22 -23 -7 -10 15 | |
      +-----------------------------------------------------------------------------------------+
      || 43 1 -41 23 -42 -14 37 -50 -32 -20 -5 -49 -9 32 -18 -22 24 43 -18 30 39 27 -30 -32 |   |
      +-----------------------------------------------------------------------------------------+
      || 12 8 26 15 22 12 0 -5 6 17 -21 -18 -33 -49 -19 33 -20 0 -15 -48 39 0 44 -19 |          |
      +-----------------------------------------------------------------------------------------+
      || -48 -27 -8 -33 -35 -16 -31 -44 -46 -49 -21 -3 -26 13 -40 4 -11 -48 36 -39 9 -39 -8 22 ||
      +-----------------------------------------------------------------------------------------+
      || 34 29 30 -1 34 -22 32 -6 39 -28 -15 37 41 -30 47 -22 -6 -7 -8 43 36 -3 35 16 |         |
      +-----------------------------------------------------------------------------------------+
      || 10 -5 -38 -25 -21 21 14 42 19 -41 8 -35 25 -31 29 3 -49 38 -35 -9 6 40 -13 -2 |        |
      +-----------------------------------------------------------------------------------------+
i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)

      +----------------------------------------------------------------------------------------+
o14 = || 1 4 44 1 40 20 29 -26 42 -16 -4 50 -40 48 23 -34 -35 37 30 4 -47 27 -31 0 |           |
      +----------------------------------------------------------------------------------------+
      || -29 -11 2 28 -12 -21 -30 20 -1 -42 27 -45 -48 -14 -3 -45 -37 30 -31 -39 -48 -29 47 0 ||
      +----------------------------------------------------------------------------------------+
      || -20 14 48 3 14 33 18 -46 24 45 -1 -29 1 31 -44 -2 -22 40 28 -49 -18 46 10 0 |         |
      +----------------------------------------------------------------------------------------+
      || 21 25 -23 -45 -48 -40 8 -7 -6 -1 14 3 -13 -44 -44 21 -41 3 30 7 13 -17 8 0 |          |
      +----------------------------------------------------------------------------------------+
      || -15 26 -17 12 0 -5 35 -11 0 -31 -39 40 49 -21 -42 28 42 -18 -29 8 30 -46 23 0 |       |
      +----------------------------------------------------------------------------------------+
      || -17 38 -49 -9 2 -1 -9 2 3 49 33 49 -46 25 1 0 -18 12 15 -28 18 -16 27 0 |             |
      +----------------------------------------------------------------------------------------+
      || -50 3 -12 -21 -10 4 36 -14 19 -9 43 -48 44 -35 16 6 20 -39 23 -21 -37 -23 19 0 |      |
      +----------------------------------------------------------------------------------------+
      || 36 -17 -27 37 -4 12 42 -16 -15 12 -3 -14 -28 38 -24 18 -9 6 -47 0 -28 47 -33 0 |      |
      +----------------------------------------------------------------------------------------+
      || 43 44 31 14 -9 -4 -4 -40 -48 45 47 17 -37 29 -9 10 -28 -33 -29 28 26 5 42 0 |         |
      +----------------------------------------------------------------------------------------+
      || 17 -47 -23 8 -18 -45 -25 2 44 41 -19 -37 5 16 2 38 -13 -20 30 44 4 22 -29 0 |         |
      +----------------------------------------------------------------------------------------+

Caveat

This routine expects the input to represent an irreducible variety

See also

Ways to use this method: