jacobian(Ideal) -- the Jacobian matrix of the generators of an ideal

Synopsis

Function: jacobian
Usage:

jacobian I
Inputs:
- I, an ideal, in a polynomial ring
Outputs:
- a matrix, the Jacobian matrix of partial derivatives of the generators of I

This is identical to jacobian generators I. See jacobian(Matrix) for more information.

i1 : R = QQ[x,y,z];

i2 : I = ideal(y^2-x*(x-1)*(x-13))

              3      2    2
o2 = ideal(- x  + 14x  + y  - 13x)

o2 : Ideal of R

i3 : jacobian I

o3 = {1} | -3x2+28x-13 |
     {1} | 2y          |
     {1} | 0           |

             3      1
o3 : Matrix R  <-- R

If the ring of I is a polynomial ring over a polynomial ring, then indeterminates in the coefficient ring are treated as constants.

i4 : R = ZZ[a,b,c][x,y,z]

o4 = R

o4 : PolynomialRing

i5 : jacobian ideal(a*y*z+b*x*z+c*x*y)

o5 = {1, 0} | cy+bz |
     {1, 0} | cx+az |
     {1, 0} | bx+ay |

             3      1
o5 : Matrix R  <-- R