
C.4 Characteristic sets
Let be the lexicographical ordering on
with
.
For let lvar() (the leading variable of ) be the largest
variable in ,
i.e., if
for some
then lvar.
Moreover, let
ini
. The pseudoremainder
of with respect to is
defined by the equality
with
and
minimal.
A set
is called triangular if
. Moreover, let ,
then is called a triangular system, if is a triangular set
such that does not vanish on
.
is called irreducible if for every there are no
,, such that
Furthermore, is called irreducible if is irreducible.
The main result on triangular sets is the following: Let
, then there are irreducible triangular sets
such that
where
. Such a set
is called an irreducible characteristic series of
the ideal .
Example:
 ring R= 0,(x,y,z,u),dp;
ideal i=3zu+y22x+2,
3x2u4yz6xz+2y2+3xy,
3z2uxu+y2z+y;
print(char_series(i));
==> _[1,1],3x2zy2+2yz,3x2u3xy2y2+2yu,
==> x, y+2z, 2y2+3yu4

