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

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

Caveat

This routine expects the input to represent an irreducible variety

See also

Ways to use this method: