# Singular

## 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