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5.1.20 division
Syntax:
division ( ideal_expression, ideal_expression )
division ( module_expression, module_expression )
Type:
- list
Purpose:
- computes the transformation matrix which expresses the generators of a
module
sm (the first argument) in terms of the generators of a
module m (the second argument) and a rest R .
Uses different algorithms depending on whether the modules are, resp. are not,
represented by standard bases.
Returns the list T,U,R , where T,U are matrices and R
is a module (resp. ideal) such that
matrix(sm)*U+matrix(R) = matrix(m)*T
and module(sm*U+matrix(R)) = module(matrix(m)*T)
(resp. ideal(sm*U+matrix(R)) = ideal(matrix(m)*T) ).
U is a diagonal matrix of units (if the basering is a polynomial ring
then U is the identity matrix).
T is the transformation matrix as returned by lift .
Example:
| ring r;
ideal m=3x2+yz,7y6+2x2y+5xz;
poly f=y7+x3+xyz+z2;
ideal i=jacob(f)+x;
division(m,i);
==> [1]:
==> _[1,1]=0
==> _[1,2]=0
==> _[2,1]=0
==> _[2,2]=1
==> _[3,1]=-16001y
==> _[3,2]=-15999x
==> _[4,1]=16001y2+3x
==> _[4,2]=16001xy-z
==> [2]:
==> _[1,1]=1
==> _[1,2]=0
==> _[2,1]=0
==> _[2,2]=1
==> [3]:
==> _[1]=0
==> _[2]=0
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See
ideal;
lift;
module.
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