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C.4 Characteristic sets

Let $<$ be the lexicographical ordering on $R=K[x_1,...,x_n]$ with $x_1 < ... < x_n$. For $f \in R$ let lvar($f$) (the leading variable of $f$) be the largest variable in $f$, i.e., if $f=a_s(x_1,...,x_{k-1})x_k^s+...+a_0(x_1,...,x_{k-1})$ for some $k \leq n$ then lvar$(f)=x_k$.

Moreover, let ini $(f):=a_s(x_1,...,x_{k-1})$. The pseudoremainder $r=\hbox{prem}(g,f)$ of $g$ with respect to $f$ is defined by the equality $\hbox{ini}(f)^a\cdot g = qf+r$ with $\hbox{deg}_{lvar(f)}(r)<\hbox{deg}_{lvar(f)}(f)$ and $a$ minimal.

A set $T=\{f_1,...,f_r\} \subset R$ is called triangular if $\hbox{lvar}(f_1)<...<\hbox{lvar}(f_r)$. Moreover, let $ U \subset T $, then $(T,U)$ is called a triangular system, if $T$ is a triangular set such that $\hbox{ini}(T)$ does not vanish on $V(T) \setminus V(U) (=:V(T\setminus U))$.

$T$ is called irreducible if for every $i$ there are no $d_i$,$f_i'$,$f_i''$ such that

\begin{displaymath}\hbox{lvar}(d_i)<\hbox{lvar}(f_i) = \hbox{lvar}(f_i')=\hbox{lvar}(f_i''),\end{displaymath}


\begin{displaymath}0 \not\in \hbox{prem}(\{ d_i, \hbox{ini}(f_i'), \hbox{ini}(f_i'')\},\{ f_1,...,f_{i-1}\}),\end{displaymath}


\begin{displaymath}\hbox{prem}(d_if_i-f_i'f_i'',\{f_1,...,f_{i-1}\})=0.\end{displaymath}

Furthermore, $(T,U)$ is called irreducible if $T$ is irreducible.

The main result on triangular sets is the following: Let $G=\{g_1,...,g_s\} \subset R$, then there are irreducible triangular sets $T_1,...,T_l$ such that $V(G)=\bigcup_{i=1}^{l}(V(T_i\setminus I_i))$ where $I_i=\{\hbox{ini}(f) \mid f \in T_i \}$. Such a set $\{T_1,...,T_l\}$ 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