next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
MultipolynomialResultants :: Discriminant

Discriminant -- resultant of the partial derivatives

Synopsis

Description

i1 : (ZZ[a_0, a_1, a_2])[x_0, x_1]; F=a_0*x_0^2+a_1*x_0*x_1+a_2*x_1^2

        2               2
o2 = a x  + a x x  + a x
      0 0    1 0 1    2 1

o2 : ZZ[a , a , a ][x , x ]
         0   1   2   0   1
i3 : time Discriminant F
     -- used 0.0112861 seconds

        2
o3 = - a  + 4a a
        1     0 2

o3 : ZZ[a , a , a ]
         0   1   2
i4 : (ZZ[a_0, a_1, a_2, a_3])[x_0, x_1]; F=a_0*x_0^3+a_1*x_0^2*x_1+a_2*x_0*x_1^2+a_3*x_1^3

        3      2          2      3
o5 = a x  + a x x  + a x x  + a x
      0 0    1 0 1    2 0 1    3 1

o5 : ZZ[a , a , a , a ][x , x ]
         0   1   2   3   0   1
i6 : time Discriminant F
     -- used 0.0125578 seconds

         2 2        3      3                     2 2
o6 = - 3a a  + 12a a  + 12a a  - 54a a a a  + 81a a
         1 2      0 2      1 3      0 1 2 3      0 3

o6 : ZZ[a , a , a , a ]
         0   1   2   3

The next example illustrates how computing the intersection of a pencil generated by two degree d forms F(x0,...,xn), G(x0,...,xn) with the discriminant hypersurface in the space of forms of degree d on n

i7 : R=ZZ/331[x_0..x_3];
i8 : F=x_0^4+x_1^4+x_2^4+x_3^4

      4    4    4    4
o8 = x  + x  + x  + x
      0    1    2    3

o8 : R
i9 : G=x_0^4-x_0*x_1^3-x_2^4+x_2*x_3^3

      4      3    4      3
o9 = x  - x x  - x  + x x
      0    0 1    2    2 3

o9 : R
i10 : R'=ZZ/331[t_0,t_1][x_0..x_3];
i11 : pencil=t_0*sub(F,R')+t_1*sub(G,R')

                4        3      4             4        3      4
o11 = (t  + t )x  - t x x  + t x  + (t  - t )x  + t x x  + t x
        0    1  0    1 0 1    0 1     0    1  2    1 2 3    0 3

o11 : R'
i12 : time D=Discriminant pencil
     -- used 2.94415 seconds

           108      106 2       102 6      100 8       98 10       96 12  
o12 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
           0        0   1       0   1      0   1       0  1        0  1   
      -----------------------------------------------------------------------
          94 14       92 16      90 18      88 20      86 22       84 24  
      161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
          0  1        0  1       0  1       0  1       0  1        0  1   
      -----------------------------------------------------------------------
        82 26     80 28      78 30       76 32      74 34       72 36  
      5t  t   - 4t  t   + 75t  t   + 103t  t   + 47t  t   + 108t  t   -
        0  1      0  1       0  1        0  1       0  1        0  1   
      -----------------------------------------------------------------------
         70 38      68 40       66 42      64 44      62 46       60 48  
      62t  t   - 97t  t   - 131t  t   + 71t  t   - 68t  t   - 144t  t   -
         0  1       0  1        0  1       0  1       0  1        0  1   
      -----------------------------------------------------------------------
          58 50      56 52      54 54       52 56     50 58      48 60  
      163t  t   + 10t  t   - 35t  t   + 105t  t   + 7t  t   + 10t  t   -
          0  1       0  1       0  1        0  1      0  1       0  1   
      -----------------------------------------------------------------------
        46 62      44 64       42 66      40 68       38 70      36 72  
      3t  t   + 76t  t   - 152t  t   - 81t  t   + 106t  t   - 11t  t   -
        0  1       0  1        0  1       0  1        0  1       0  1   
      -----------------------------------------------------------------------
         34 74      32 76      30 78      28 80     26 82      24 84  
      13t  t   + 17t  t   + 18t  t   + 88t  t   + 9t  t   + 58t  t   -
         0  1       0  1       0  1       0  1      0  1       0  1   
      -----------------------------------------------------------------------
         22 86       20 88       18 90       16 92       14 94      12 96  
      73t  t   + 113t  t   - 154t  t   - 102t  t   - 161t  t   + 33t  t   -
         0  1        0  1        0  1        0  1        0  1       0  1   
      -----------------------------------------------------------------------
          10 98      8 100       6 102       4 104      2 106      108
      130t  t   - 21t t    + 157t t    + 105t t    + 82t t    + 69t
          0  1       0 1         0 1         0 1        0 1        1

       ZZ
o12 : ---[t , t ]
      331  0   1
i13 : factor D

                  9            9           9           9         18         18    2              2 9    2              2 9
o13 = (128t  - t ) (128t  + t ) (11t  - t ) (11t  + t ) (t  - t )  (t  + t )  (39t  - 139t t  + t ) (39t  + 139t t  + t ) (69)
           0    1       0    1      0    1      0    1    0    1     0    1       0       0 1    1      0       0 1    1

o13 : Expression of class Product

See also

Ways to use Discriminant :