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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 = {| -25 20 -30 -16 24 -36 |, | 19 -29 19 23 -29 19 |, | -44 46 -8 7 -10
     ------------------------------------------------------------------------
     -29 |, | 8 41 -24 46 -22 -29 |}

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

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

Caveat

This routine expects the input to represent an irreducible variety

See also