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Resultants :: discriminant

discriminant -- resultant of the partial derivatives

Synopsis

Description

The discriminant of a homogeneous polynomial is defined, up to a scalar factor, as the resultant of its partial derivatives. For the general theory, see one of the following: Using Algebraic Geometry, by David A. Cox, John Little, Donal O'shea; Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky.

i1 : ZZ[a,b,c][x,y]; F = a*x^2+b*x*y+c*y^2

        2              2
o2 = a*x  + b*x*y + c*y

o2 : ZZ[a..c][x..y]
i3 : time discriminant F
     -- used 0.00571206 seconds

        2
o3 = - b  + 4a*c

o3 : ZZ[a..c]
i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3

        3      2         2      3
o5 = a*x  + b*x y + c*x*y  + d*y

o5 : ZZ[a..d][x..y]
i6 : time discriminant F
     -- used 0.00780499 seconds

        2 2       3     3                   2 2
o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d

o6 : ZZ[a..d]

The next example illustrates how computing the intersection of a pencil generated by two degree $d$ forms $F(x_0,\ldots,x_n), G(x_0,\ldots,x_n)$ with the discriminant hypersurface in the space of forms of degree $d$ on $\mathbb{P}^n$

i7 : x=symbol x; R=ZZ/331[x_0..x_3]

o8 = R

o8 : PolynomialRing
i9 : F=x_0^4+x_1^4+x_2^4+x_3^4

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

o9 : R
i10 : G=x_0^4-x_0*x_1^3-x_2^4+x_2*x_3^3

       4      3    4      3
o10 = x  - x x  - x  + x x
       0    0 1    2    2 3

o10 : R
i11 : R'=ZZ/331[t_0,t_1][x_0..x_3];
i12 : pencil=t_0*sub(F,R')+t_1*sub(G,R')

                4        3      4             4        3      4
o12 = (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

o12 : R'
i13 : time D=discriminant pencil
     -- used 0.388131 seconds

           108      106 2       102 6      100 8       98 10       96 12  
o13 = - 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
o13 : ---[t ..t ]
      331  0   1
i14 : factor D

                  9            9           9           9         18         18    2              2 9    2              2 9
o14 = (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

o14 : Expression of class Product

See also

Ways to use discriminant :

For the programmer

The object discriminant is a method function with options.