Singular Manual: Characteristic sets

  | ## 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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