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

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

Caveat

This routine expects the input to represent an irreducible variety

See also

Ways to use this method: