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5.1.66 jacob

Syntax:
jacob ( poly_expression )
jacob ( ideal_expression )
jacob ( module_expression )
Type:
ideal, if the input is a polynomial
matrix, if the input is an ideal
module, if the input is a module
Purpose:
computes the Jacobi ideal, resp. Jacobi matrix, generated by all partial derivatives of the input.
Note:
In a ring with n variables, jacob of a module or an ideal (considered as matrix with a single a row) or a polynomial (considered as a matrix with a single entry) is the matrix consisting of horizontally concatenated blocks (in this order): diff(MT,var(1)), ... , diff(MT,var(n)), where MT is the transposed input argument considered as a matrix.
Example:
 
  ring R;
  poly f = x2yz + xy3z + xyz5;
  ideal i = jacob(f); i;
==> i[1]=yz5+y3z+2xyz
==> i[2]=xz5+3xy2z+x2z
==> i[3]=5xyz4+xy3+x2y
  matrix m = jacob(i);
  print(m);
==> 2yz,        z5+3y2z+2xz, 5yz4+y3+2xy, 
==> z5+3y2z+2xz,6xyz,        5xz4+3xy2+x2,
==> 5yz4+y3+2xy,5xz4+3xy2+x2,20xyz3       
  print(jacob(m));
==> 0, 2z,        2y,        2z,        6yz,5z4+3y2+2x,2y,        5z4+3y2+2x,\
   20yz3,
==> 2z,6yz,       5z4+3y2+2x,6yz,       6xz,6xy,       5z4+3y2+2x,6xy,       \
   20xz3,
==> 2y,5z4+3y2+2x,20yz3,     5z4+3y2+2x,6xy,20xz3,     20yz3,     20xz3,     \
   60xyz2
See diff; ideal; module; nvars.

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