Description
The discriminant of a homogeneous polynomial is defined, up to a scalar factor, as the resultant(Matrix) 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]
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i3 : time discriminant F
-- used 0.00692138 seconds
2
o3 = - b + 4a*c
o3 : ZZ[a..c]
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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]
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i6 : time discriminant F
-- used 0.0076513 seconds
2 2 3 3 2 2
o6 = - b c + 4a*c + 4b d - 18a*b*c*d + 27a d
o6 : ZZ[a..d]
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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
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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
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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
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i11 : R'=ZZ/331[t_0,t_1][x_0..x_3];
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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'
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i13 : time D=discriminant pencil
-- used 0.33765 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
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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
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