C.1 Standard bases
==================

Definition
----------
Let $R = \hbox{Loc}_< K[\underline{x}]$ and let $I$ be a submodule of $R^r$.
Note that for r=1 this means that $I$ is an ideal in $R$.
Denote by $L(I)$ the submodule of $R^r$ generated by the leading terms 
of elements of $I$, i.e. by $\left\{L(f) \mid f \in I\right\}$.
Then $f_1, \ldots, f_s \in I$ is called a {\bf standard basis} of $I$ 
if $L(f_1), \ldots, L(f_s)$ generate $L(I)$.

Properties
----------
normal form:
A function $\hbox{NF} : R^r \times \{G \mid G\ \hbox{ a standard
basis}\} \to R^r, (p,G) \mapsto \hbox{NF}(p|G)$, is called a {\bf normal
form} if for any $p \in R^r$ and any standard basis $G$ the following
holds: if $\hbox{NF}(p|G) \not= 0$ then $L(g)$ does not divide
$L(\hbox{NF}(p|G))$ for all $g \in G$.

\noindent
$\hbox{NF}(p|G)$ is called a {\bf normal form of} $p$ {\bf with
respect to} $G$ (note that such a function is not unique).
ideal membership:
For a standard basis $G$ of $I$ the following holds: 
$f \in I$ if and only if $\hbox{NF}(f,G) = 0$.
Hilbert function:
Let \hbox{$I \subseteq K[\underline{x}]^r$} be a homogeneous module, then the Hilbert function
$H_I$ of $I$ (see below)
and the Hilbert function $H_{L(I)}$ of the leading module $L(I)$
coincide, i.e.,
$H_I=H_{L(I)}$.

<font size="-1">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; User manual for <a href="http://www.singular.uni-kl.de/"><i>Singular</i></a> version 2-0-4, October 2002,
generated by <a href="http://www.gnu.org/software/texinfo/"><i>texi2html</i></a>.
</font>

</body>
</html>
