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C.4 Characteristic sets
Let
< be the lexicographical ordering on R = K[x1,...,xn] with x1 < ... < xn. For f ∈ R let lvar(f) (the leading variable
of f) be the largest variable in f, i.e., if f = as(x1,...,xk−1)xks + ... + a0(x1,...,xk−1) for some k ≤ n then
lvar(f) = xk.
Moreover, let ini(f) := as(x1,...,xk−1). The pseudo remainder r = prem(g,f) of g with respect to f is defined
by the equality ini(f)a ⋅ g = qf + r with deglvar(f)(r) < deglvar(f)(f) and a minimal.
A set T = {f1,...,fr}⊂ R is called triangular if lvar(f1) < ... < lvar(fr). Moreover, let U ⊂ T,
then (T,U) is called a triangular system, if T is a triangular set such that ini(T) does not vanish on
V (T) \ V (U)(=: V (T \ U)).
T is called irreducible if for every i there are no di,fi‘,fi“ such that
Furthermore, (T,U) is called irreducible if T is irreducible.
The main result on triangular sets is the following: let G = {g1,...,gs}⊂ R then there are irreducible
triangular sets T1,...,Tl such that V (G) = ⋃
i=1l(V (Ti \ Ii)) where Ii = {ini(f)∣f ∈ Ti}. Such a set
{T1,...,Tl} is called an irreducible characteristic series of the ideal (G).
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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