The elements f and g should be polynomials in the same ring, and x should be a variable in that ring. The result is the determinant of the Sylvester matrix, sylvesterMatrix(f,g,x). The resultant of f and its derivative with respect to x is the discriminant, discriminant(f,x).
i1 : R = ZZ[x,a,b,c,d] o1 = R o1 : PolynomialRing |
i2 : f = x^7+3*x^4+a*x+b
7 4
o2 = x + 3x + x*a + b
o2 : R
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i3 : g = x^8+x^5+c*x+d
8 5
o3 = x + x + x*c + d
o3 : R
|
i4 : time eliminate(ideal(f,g),x)
-- used 1.07207 seconds
7 8 3 5 8 6 3 4 7 3 3 2
o4 = ideal(a b*c - a d + a b - b - 6a b*c - 18a b c + 7b c + 48a b c -
------------------------------------------------------------------------
6 2 3 2 3 5 3 3 4 4 4 3 5 2 6 7
21b c - 46a b c + 35b c + 15a b*c - 35b c + 21b c - 7b c + b*c +
------------------------------------------------------------------------
7 4 3 6 4 2 5 4 2 4 2
6a d + 21a b d - 8a*b d - 85a b c*d + 41a*b c*d + 79a b*c d - 85a*b c d
------------------------------------------------------------------------
4 3 3 3 2 4 5 6 5 2
- 15a c d + 90a*b c d - 50a*b c d + 13a*b*c d - a*c d + 37a b*d -
------------------------------------------------------------------------
2 4 2 5 2 2 3 2 2 2 2 2 2 3 2 2 4 2
20a b d - 33a c*d + 66a b c*d - 78a b c d + 38a b*c d - 6a c d -
------------------------------------------------------------------------
3 2 3 3 3 3 2 3 4 4 2 5 5 2 4
16a b d + 25a b*c*d - 9a c d - 2a d - 6a b + 12a b*c + 84a b c -
------------------------------------------------------------------------
2 3 2 2 2 3 2 4 6 3 3 6
328a b c + 428a b c - 178a b*c - 12a d - 102a b d - 10b d +
------------------------------------------------------------------------
3 2 5 3 2 4 2 3 3 3 3
578a b c*d + 22b c*d - 710a b*c d + 12b c d + 178a c d - 68b c d +
------------------------------------------------------------------------
2 4 5 4 2 4 2 4 2 3 2
62b c d - 18b*c d - 250a b*d - 2a*b d + 282a c*d - 40a*b c*d +
------------------------------------------------------------------------
2 2 2 3 2 4 2 2 2 3 2 3 2 2 3
104a*b c d - 80a*b*c d + 18a*c d + 4a b d - 78a b*c*d + 18a c d +
------------------------------------------------------------------------
3 4 3 4 2 4 2 4 3 4 5 5 7
87a d - 17b d + 37b c*d - 23b*c d + 3c d - 35a*b*d + 21a*c*d - d
------------------------------------------------------------------------
5 4 4 3 2 2 3 4 5
+ 12a*b - 8a b*c - 152a*b c + 656a*b c - 1128a*b c + 612a*b*c + 8a d
------------------------------------------------------------------------
2 3 2 2 2 2 2 3 3 2 4 2
+ 188a b d - 1188a b c*d + 1884a b*c d - 612a c d + 532a b*d - 88b d -
------------------------------------------------------------------------
3 2 3 2 2 2 2 2 3 3 2 4
756a c*d + 64b c*d + 24b c d + 196a*b d + 156a*b*c*d - 714a d +
------------------------------------------------------------------------
5 5 5 4 3 2 2 3 4 3
114b*d - 54c*d - 8b + 96b c - 432b c + 864b c - 648b*c - 120a*b d
------------------------------------------------------------------------
2 2 3 2 2 2 2 2 3
+ 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d + 648a c*d - 504b d
------------------------------------------------------------------------
3 4 4
- 216b*c*d + 2052a*d - 1944d )
o4 : Ideal of R
|
i5 : time ideal resultant(f,g,x)
-- used 0.032002 seconds
7 8 3 5 8 6 3 4 7 3 3 2
o5 = ideal(- a b*c + a d - a b + b + 6a b*c + 18a b c - 7b c - 48a b c +
------------------------------------------------------------------------
6 2 3 2 3 5 3 3 4 4 4 3 5 2 6 7
21b c + 46a b c - 35b c - 15a b*c + 35b c - 21b c + 7b c - b*c -
------------------------------------------------------------------------
7 4 3 6 4 2 5 4 2 4 2
6a d - 21a b d + 8a*b d + 85a b c*d - 41a*b c*d - 79a b*c d + 85a*b c d
------------------------------------------------------------------------
4 3 3 3 2 4 5 6 5 2
+ 15a c d - 90a*b c d + 50a*b c d - 13a*b*c d + a*c d - 37a b*d +
------------------------------------------------------------------------
2 4 2 5 2 2 3 2 2 2 2 2 2 3 2 2 4 2
20a b d + 33a c*d - 66a b c*d + 78a b c d - 38a b*c d + 6a c d +
------------------------------------------------------------------------
3 2 3 3 3 3 2 3 4 4 2 5 5 2 4
16a b d - 25a b*c*d + 9a c d + 2a d + 6a b - 12a b*c - 84a b c +
------------------------------------------------------------------------
2 3 2 2 2 3 2 4 6 3 3 6
328a b c - 428a b c + 178a b*c + 12a d + 102a b d + 10b d -
------------------------------------------------------------------------
3 2 5 3 2 4 2 3 3 3 3
578a b c*d - 22b c*d + 710a b*c d - 12b c d - 178a c d + 68b c d -
------------------------------------------------------------------------
2 4 5 4 2 4 2 4 2 3 2
62b c d + 18b*c d + 250a b*d + 2a*b d - 282a c*d + 40a*b c*d -
------------------------------------------------------------------------
2 2 2 3 2 4 2 2 2 3 2 3 2 2 3
104a*b c d + 80a*b*c d - 18a*c d - 4a b d + 78a b*c*d - 18a c d -
------------------------------------------------------------------------
3 4 3 4 2 4 2 4 3 4 5 5 7
87a d + 17b d - 37b c*d + 23b*c d - 3c d + 35a*b*d - 21a*c*d + d
------------------------------------------------------------------------
5 4 4 3 2 2 3 4 5
- 12a*b + 8a b*c + 152a*b c - 656a*b c + 1128a*b c - 612a*b*c - 8a d
------------------------------------------------------------------------
2 3 2 2 2 2 2 3 3 2 4 2
- 188a b d + 1188a b c*d - 1884a b*c d + 612a c d - 532a b*d + 88b d +
------------------------------------------------------------------------
3 2 3 2 2 2 2 2 3 3 2 4
756a c*d - 64b c*d - 24b c d - 196a*b d - 156a*b*c*d + 714a d -
------------------------------------------------------------------------
5 5 5 4 3 2 2 3 4 3
114b*d + 54c*d + 8b - 96b c + 432b c - 864b c + 648b*c + 120a*b d
------------------------------------------------------------------------
2 2 3 2 2 2 2 2 3
- 792a*b c*d + 1512a*b*c d - 648a*c d + 360a b*d - 648a c*d + 504b d
------------------------------------------------------------------------
3 4 4
+ 216b*c*d - 2052a*d + 1944d )
o5 : Ideal of R
|
i6 : sylvesterMatrix(f,g,x)
o6 = {0} | b 0 0 0 0 0 0 0 d 0 0 0 0 0 0 |
{1} | a b 0 0 0 0 0 0 c d 0 0 0 0 0 |
{2} | 0 a b 0 0 0 0 0 0 c d 0 0 0 0 |
{3} | 0 0 a b 0 0 0 0 0 0 c d 0 0 0 |
{4} | 3 0 0 a b 0 0 0 0 0 0 c d 0 0 |
{5} | 0 3 0 0 a b 0 0 1 0 0 0 c d 0 |
{6} | 0 0 3 0 0 a b 0 0 1 0 0 0 c d |
{7} | 1 0 0 3 0 0 a b 0 0 1 0 0 0 c |
{8} | 0 1 0 0 3 0 0 a 1 0 0 1 0 0 0 |
{9} | 0 0 1 0 0 3 0 0 0 1 0 0 1 0 0 |
{10} | 0 0 0 1 0 0 3 0 0 0 1 0 0 1 0 |
{11} | 0 0 0 0 1 0 0 3 0 0 0 1 0 0 1 |
{12} | 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 |
{13} | 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 |
{14} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
15 15
o6 : Matrix R <--- R
|
i7 : discriminant(f,x)
7 6 3 3 6 5 2 3
o7 = 46656a - 314928a - 1555848a b + 823543b + 708588a + 9501786a b -
------------------------------------------------------------------------
4 3 3
531441a - 20575296a*b + 15116544b
o7 : R
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