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

      +-------------------------------------------------------------------------------------+
o14 = || -38 23 -11 16 37 24 -35 14 -42 27 -31 -19 37 -7 -42 -33 -35 -31 -40 -47 27 30 4 0 ||
      +-------------------------------------------------------------------------------------+
      || 42 -19 17 -24 -26 -22 -36 22 1 26 22 13 30 -35 44 28 -37 47 -48 -48 -29 -31 -39 0 ||
      +-------------------------------------------------------------------------------------+
      || -27 23 -7 16 19 18 47 20 47 50 -46 1 40 11 10 -30 -22 10 1 -18 46 28 -49 0 |       |
      +-------------------------------------------------------------------------------------+
      || 3 41 33 16 7 32 6 17 23 -29 -19 -43 3 -5 -23 -48 -41 8 -13 13 -17 30 7 0 |         |
      +-------------------------------------------------------------------------------------+
      || 50 -40 28 11 7 44 4 1 -14 5 32 -28 -18 24 6 30 42 23 49 30 -46 -29 8 0 |           |
      +-------------------------------------------------------------------------------------+
      || 27 47 -5 20 39 -45 -18 -31 33 34 -29 -3 12 -38 39 17 -18 27 -46 18 -16 15 -28 0 |  |
      +-------------------------------------------------------------------------------------+
      || -44 38 26 50 34 -9 31 40 12 -49 2 -21 -39 -33 42 21 20 19 44 -37 -23 23 -21 0 |    |
      +-------------------------------------------------------------------------------------+
      || -18 8 -43 -20 12 -30 43 -46 38 40 3 39 6 6 -9 -14 -9 -33 -28 -28 47 -47 0 0 |      |
      +-------------------------------------------------------------------------------------+
      || -32 47 -49 8 -50 32 40 -46 -30 -14 -17 25 -33 41 34 15 -28 42 -37 26 5 -29 28 0 |  |
      +-------------------------------------------------------------------------------------+
      || -13 -40 -37 29 -41 -17 -37 -28 -49 -14 -9 -38 -20 32 29 49 -13 -29 5 4 22 30 44 0 ||
      +-------------------------------------------------------------------------------------+

Caveat

This routine expects the input to represent an irreducible variety

See also