jacobian(Ring) -- the Jacobian matrix of the polynomials defining a quotient ring

Synopsis

Function: jacobian
Usage:

jacobian R
Inputs:
- R, a ring, a quotient of a polynomial ring
Outputs:
- a matrix, the Jacobian matrix of partial derivatives of the presentation matrix of R

This is identical to jacobian presentation R, except that the resulting matrix is over the ring R. See jacobian(Matrix) for more information.

i1 : R = QQ[x,y,z]/(y^2-x^3-x^7);

i2 : jacobian R

o2 = {1} | -7x6-3x2 |
     {1} | 2y       |
     {1} | 0        |

             3      1
o2 : Matrix R  <-- R

If the ring R is a (quotient of a) polynomial ring over a polynomial ring, then the top set of indeterminates is used, on the top set of quotients:

i3 : A = ZZ[a,b,c]/(a^2+b^2+c^2);

i4 : R = A[x,y,z]/(a*x+b*y+c*z-1)

o4 = R

o4 : QuotientRing

i5 : jacobian R

o5 = {1, 0} | a |
     {1, 0} | b |
     {1, 0} | c |

             3      1
o5 : Matrix R  <-- R