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5.1.76 liftstd

Syntax:
liftstd ( ideal_expression, matrix_name )
liftstd ( module_expression, matrix_name )
liftstd ( ideal_expression, matrix_name, module_name )
liftstd ( module_expression, matrix_name, module_name )
Type:
ideal or module
Purpose:
returns a standard basis of an ideal or module and the transformation matrix from the given ideal, resp. module, to the standard basis.
That is, if m is the ideal or module, sm the standard basis returned by liftstd, and T the transformation matrix then matrix(sm)=matrix(m)*T and sm=ideal(matrix(m)*T), resp. sm=module(matrix(m)*T).
In an optional third argument the syzygy module will be returned.
Note:
if working in a quotient ring, then matrix(sm)=reduce(matrix(m)*T,0) and sm=reduce(ideal(matrix(m)*T),0).
Example:
 
  ring R=0,(x,y,z),dp;
  poly f=x3+y7+z2+xyz;
  ideal i=jacob(f);
  matrix T;
  ideal sm=liftstd(i,T);
  sm;
==> sm[1]=xy+2z
==> sm[2]=3x2+yz
==> sm[3]=yz2+3048192z3
==> sm[4]=3024xz2-yz2
==> sm[5]=y2z-6xz
==> sm[6]=3097158156288z4+2016z3
==> sm[7]=7y6+xz
  print(T);
==> 0,1,T[1,3],   T[1,4],y,  T[1,6],0,
==> 0,0,-3x+3024z,3x,    0,  T[2,6],1,
==> 1,0,T[3,3],   T[3,4],-3x,T[3,6],0 
  matrix(sm)-matrix(i)*T;
==> _[1,1]=0
==> _[1,2]=0
==> _[1,3]=0
==> _[1,4]=0
==> _[1,5]=0
==> _[1,6]=0
==> _[1,7]=0
  module s;
  sm=liftstd(i,T,s);
  print(s);
==> -xy-2z,7y6+xz, -7056y6-1008xz,     
==> 0,     -3x2-yz,3024x2-xy+1008yz-2z,
==> 3x2+yz,0,      7y6+xz              
See ideal; matrix; option; ring; std.

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