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C.4 Characteristic sets
Let 228#228 be the lexicographical ordering on
674#674 with
675#675.
For 676#676 let lvar(267#267) (the leading variable of 267#267) be the largest
variable in 267#267,
i.e., if
677#677 for some
678#678 then lvar679#679.
Moreover, let
ini
680#680. The pseudoremainder
681#681 of 149#149 with respect to 267#267 is
defined by the equality
682#682 with
683#683 and 4#4
minimal.
A set
684#684 is called triangular if
685#685. Moreover, let 686#686,
then 687#687 is called a triangular system, if 279#279 is a triangular set
such that 688#688 does not vanish on
689#689.
279#279 is called irreducible if for every 57#57 there are no
690#690,691#691,692#692 such that
693#693
694#694
695#695
Furthermore, 687#687 is called irreducible if 279#279 is irreducible.
The main result on triangular sets is the following: Let
696#696, then there are irreducible triangular sets 697#697
such that
698#698
where
699#699. Such a set
700#700 is called an irreducible characteristic series of
the ideal 701#701.
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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