Top
Back: G-algebras
Forward: Syzygies and resolutions
FastBack: Mathematical background
FastForward: PLURAL libraries
Up: Mathematical background
Top: Plural Manual
Contents: Table of Contents
Index: Index
About: About this document

A.2 Groebner bases

Let $<$ be a fixed monomial well-ordering on the $G$-algebra $A$.

Definition

For a set $S \subset A^r$, define $\ell(S) \subseteq N^n$ to be the monoid, generated by the leading exponents of elements of $S$, that is $\ell(S)=\langle \alpha \mid \exists s \in S, \hbox{lm(s)}=x^{\alpha} \rangle
\subseteq N^n$. We call $\ell(S)$ the monoid of leading exponents. There exist $\alpha_1, \ldots, \alpha_m \in N^n$, such that $\ell(S) :=\langle \alpha_1, \ldots, \alpha_m \rangle$. We define a set of leading monomials of $S$ be $L(S) := \{ x^{\alpha} \mid \alpha \in \ell(S) \}\subset{A}$.

A finite set $G\subset I$ is called Groebner basis of $I$ if and only if $L(G)=L(I)$, that is for any $f \in I\setminus \{ 0 \}$ there exists a $g\in G$ satisfying $ \hbox{lm}(g) \mid \hbox{lm}(f)$.


Remark: In the non-commutative case we are working with well ordering only. (See PLURAL conventions)

A Groebner basis $G\subset A^r$ is called minimal if $0\notin G$ and if lm $(g)\notin L(G\setminus \{ g \})$ for all $g\in G$. Note, that any Groebner basis can be made minimal by deleting successively those $g$ with $\hbox{lm}(h)\mid \hbox{lm}(g)$ for some $h\in G\setminus\{g \}$.

For $f\in A^r $ and $G\subset A^r$ we say that $f$ is reduced with respect to $G$ if no monomial of $f$ is contained in $L(G)$.

Normal Form

A map $\hbox{NF} : A^r \times \{G \mid G\ \hbox{ a Groebner
basis}\} \to A^r, (f\vert G) \mapsto \hbox{NF}(f\vert G)$, is called a normal form on $A^r$ if for any $f\in A^r $ and any Groebner basis $G$ the following holds:

(i) $\hbox{NF}(f\vert G) \not= 0$ then $\hbox{lm}(g)$ does not divide $\hbox{lm}(\hbox{NF}(f\vert G))$ for all $g\in G$.

(ii) $f - \hbox{NF}(f\vert G)\in \langle G \rangle.$

$\hbox{NF}(f\vert G)$ is called a normal form of $f$ with respect to $G$ (note that such a map is not unique).


Remark: With respect to the definitions of ideal and module (see PLURAL conventions ) PLURAL works with left normal form only.

Ideal membership:
For a Groebner basis $G$ of $I$ the following holds: $f \in I$ if and only if $\hbox{NF}(f\vert G) = 0$.


Top Back: G-algebras Forward: Syzygies and resolutions FastBack: Mathematical background FastForward: PLURAL libraries Up: Mathematical background Top: Plural Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 2-1-99, August 2004, generated by texi2html.