next | previous | forward | backward | up | top | index | toc | Macaulay2 website
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 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18 -43 21 -38 | |
      +-------------------------------------------------------------------------------------+
      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34 -47 38 -15 ||
      +-------------------------------------------------------------------------------------+
      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17 48 7 47 |     |
      +-------------------------------------------------------------------------------------+
      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -3 -38 36 |     |
      +-------------------------------------------------------------------------------------+
      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37 -10 -7 22 |  |
      +-------------------------------------------------------------------------------------+
      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20 -30 27 30 |  |
      +-------------------------------------------------------------------------------------+
      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17 44 0 -48 |  |
      +-------------------------------------------------------------------------------------+
      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49 -8 -39 -39 | |
      +-------------------------------------------------------------------------------------+
      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35 -3 43 |      |
      +-------------------------------------------------------------------------------------+
      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41 -13 40 -9 | |
      +-------------------------------------------------------------------------------------+
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: