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RCS file: /home/cvs/OpenXM/doc/Papers/Attic/dag-noro-proc.tex,v
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--- OpenXM/doc/Papers/Attic/dag-noro-proc.tex	2001/11/30 02:08:46	1.8
+++ OpenXM/doc/Papers/Attic/dag-noro-proc.tex	2001/12/28 06:06:15	1.9
@@ -1,4 +1,4 @@
-% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.7 2001/11/30 02:02:09 noro Exp $
+% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.8 2001/11/30 02:08:46 noro Exp $
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % This is a sample input file for your contribution to a multi-
 % author book to be published by Springer Verlag.
@@ -213,11 +213,84 @@ Groebner basis computation over {\bf Q}, which are eas
 implemented but may not be well known.
 We use the following notations.
 \begin{description}
-\item $Id(F)$ : a polynomial ideal generated by a polynomial set $F$
-\item $\phi_p$ : the canonical projection from ${\bf Z}$ onto $GF(p)$
-\item $HT(f)$ : the head term of a polynomial with respect to a term order
-\item $HC(f)$ : the head coefficient of a polynomial with respect to a term order
+\item $<$ : a term order in the set of monomials. It is a total order such that
+
+ $\forall t, 1 \le t$ and $\forall s, t, u, s<t \Rightarrow us<ut$.
+\item $Id(F)$ : a polynomial ideal generated by a polynomial set $F$.
+\item $HT(f)$ : the head term of a polynomial with respect to a term order.
+\item $HC(f)$ : the head coefficient of a polynomial with respect to a term order.
+\item $T(f)$ : terms with non zero coefficients in $f$.
+\item $Spoly(f,g)$ : the S-polynomial of $\{f,g\}$ 
+
+$Spoly(f,g) = T_{f,g}/HT(f)\cdot f/HC(f) -T_{f,g}/HT(g)\cdot g/HC(g)$, where
+$T_{f,g} = LCM(HT(f),HT(g))$.
+\item $\phi_p$ : the canonical projection from ${\bf Z}$ onto $GF(p)$.
 \end{description}
+
+\subsection{Groebner basis computation and its improvements}
+
+A Groebner basis of an ideal $Id(F)$ can be computed by the Buchberger
+algorithm. The key oeration in the algorithm is the following 
+division by a polynomial set.
+\begin{tabbing}
+while \= $\exists g \in G$, $\exists t \in T(f)$ such that $HT(g)|t$ do\\
+      \> $f \leftarrow f - t/HT(g) \cdot c/HC(g) \cdot g$, \quad
+      where $c$ is the coeffcient of $t$ in $f$
+\end{tabbing}
+This division terminates for any term order.
+With this division, we can show the most primitive version of the
+Buchberger algorithm.
+\begin{tabbing}
+Input : a finite polynomial set $F$\\
+Output : a Groebner basis $G$ of $Id(F)$ with respect to a term order $<$\\
+$G \leftarrow F$; \quad $D \leftarrow \{\{f,g\}| f, g \in G, f \neq g \}$\\
+while \= $D \neq \emptyset$ do \\
+      \> $\{f,g\} \leftarrow$ an element of $D$; \quad
+          $D \leftarrow D \setminus \{P\}$\\
+      \> $R \leftarrow$ a remainder of $Spoly(f,g)$ on division by $G$\\
+      \> if $R \neq 0$ then $D \leftarrow D \cup \{\{f,R\}| f \in G\}$; \quad
+         $G \leftarrow G \cup \{R\}$\\
+end do\\
+return G
+\end{tabbing}
+Though this algorithm gives a Groebner basis of $Id(F)$, 
+it is not practical at all. We need lots of techniques to make
+it practical. The following are major improvements:
+\begin{itemize}
+\item Useless pair detection
+
+We don't have to process all the pairs in $D$ and several useful
+criteria for detecting useless pairs were proposed.
+
+\item Selection strategy
+
+The selection of $\{f,g\}$ greatly affects the subsequent computation.
+The typical strategies are the normal startegy and the sugar strategy.
+The latter was proposed for efficient computation under a non 
+degree-compatible order.
+
+\item Modular methods
+
+Even if we apply several criteria, it is difficult to detect all pairs
+whose S-polynomials are reduced to zero, and the cost to process them
+often occupies a major part in the whole computation. The trace algorithms
+were proposed to reduce such cost. This will be explained in more detail
+in Section \ref{gbhomo}.
+
+\item Change of ordering
+
+For elimination, we need a Groebner basis with respect to a non
+degree-compatible order, but it is often hard to compute it by
+the Buchberger algorithm. If the ideal is zero dimensional, we
+can apply a change of ordering algorithm for a Groebner basis
+with respect to any order and we can obtain a Groebner basis
+with respect to a desired order.
+
+\end{itemize}
+By implementing these techniques, one can obtain Groebner bases for
+wider range of inputs. Nevertheless there are still intractable
+problems with these classical tools. In the subsequent sections
+we show several methods for further improvements.
 
 \subsection{Combination of homogenization and trace lifting}
 \label{gbhomo}