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GroebnerStrata :: randomPointsOnRationalVariety(Ideal,ZZ)

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

      +-------------------------------------------------------------------------------------+
o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
      +-------------------------------------------------------------------------------------+
      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
      +-------------------------------------------------------------------------------------+
      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
      +-------------------------------------------------------------------------------------+
      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
      +-------------------------------------------------------------------------------------+
      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
      +-------------------------------------------------------------------------------------+
      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
      +-------------------------------------------------------------------------------------+
      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
      +-------------------------------------------------------------------------------------+
      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
      +-------------------------------------------------------------------------------------+
      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
      +-------------------------------------------------------------------------------------+
      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
      +-------------------------------------------------------------------------------------+

Caveat

This routine expects the input to represent an irreducible variety

See also