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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 standard basis of $I$ if $L(f_1), \ldots, L(f_s)$ generate $L(I)$.

A standard basis is minimal if $\forall i: (f_1,..,f_{i-1},f_{i+1},..,f_s) \neq I$.

A minimal standard basis is completely reduced if $\forall i: {\tt reduce}(f_i,(f_1,..,f_{i-1},f_{i+1},..,f_s))=f_i$

Properties

normal form:
A function $\hbox{NF} : R^r \times \{G \mid G\ \hbox{ a standard basis}\} \to R^r, (p,G) \mapsto \hbox{NF}(p\vert G)$, is called a normal form if for any $p \in R^r$ and any standard basis $G$ the following holds: if $\hbox{NF}(p\vert G) \not= 0$ then $L(g)$ does not divide $L(\hbox{NF}(p\vert G))$ for all $g\in G$. The function may also be applied to any generating set of an ideal: the result is then not uniquely defined.

$\hbox{NF}(p\vert G)$ is called a normal form of $p$ with respect to $G$

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 $I \subseteq K[\underline{x}]^r$ 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., $H_I=H_{L(I)}$.