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 % This is a sample input file for your contribution to a multi-  % This is a sample input file for your contribution to a multi-
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Line 213  Groebner basis computation over {\bf Q}, which are eas
Line 213  Groebner basis computation over {\bf Q}, which are eas
 implemented but may not be well known.  implemented but may not be well known.
 We use the following notations.  We use the following notations.
 \begin{description}  \begin{description}
 \item $Id(F)$ : a polynomial ideal generated by a polynomial set $F$  \item $<$ : a term order in the set of monomials. It is a total order such that
 \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   $\forall t, 1 \le t$ and $\forall s, t, u, s<t \Rightarrow us<ut$.
 \item $HC(f)$ : the head coefficient of a polynomial with respect to a term order  \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}  \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}  \subsection{Combination of homogenization and trace lifting}
 \label{gbhomo}  \label{gbhomo}
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