## 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+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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