Singular Manual: Characteristic sets

  | ## 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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