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C.4 Characteristic sets
Let 226#226 be the lexicographical ordering on
672#672 with
673#673.
For 674#674 let lvar(265#265) (the leading variable of 265#265) be the largest
variable in 265#265,
i.e., if
675#675 for some
676#676 then lvar677#677.
Moreover, let
ini
678#678. The pseudoremainder
679#679 of 149#149 with respect to 265#265 is
defined by the equality
680#680 with
681#681 and 4#4
minimal.
A set
682#682 is called triangular if
683#683. Moreover, let 684#684,
then 685#685 is called a triangular system, if 277#277 is a triangular set
such that 686#686 does not vanish on
687#687.
277#277 is called irreducible if for every 57#57 there are no
688#688,689#689,690#690 such that
691#691
692#692
693#693
Furthermore, 685#685 is called irreducible if 277#277 is irreducible.
The main result on triangular sets is the following: Let
694#694, then there are irreducible triangular sets 695#695
such that
696#696
where
697#697. Such a set
698#698 is called an irreducible characteristic series of
the ideal 699#699.
Example:
| ring R= 0,(x,y,z,u),dp;
ideal i=-3zu+y2-2x+2,
-3x2u-4yz-6xz+2y2+3xy,
-3z2u-xu+y2z+y;
print(char_series(i));
==> _[1,1],3x2z-y2+2yz,3x2u-3xy-2y2+2yu,
==> x, -y+2z, -2y2+3yu-4
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