# 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:
• , the Jacobian matrix of partial derivatives of the presentation matrix of R

## Description

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

## Ways to use this method:

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